Spoilers for Round 921, Div. 1D. Plot written by me.

**Scene 1**

**Scene 1: A Fateful Encounter**

*[The university campus is bustling with students moving between classes, their voices blending into a steady hum of activity. Underneath a large oak tree in a quieter corner of the courtyard, Kai sits with his laptop, his brow furrowed in concentration as he tackles a challenging coding problem.]*

**Narrator:** (voiceover) "In the heart of the bustling campus, where minds collided and ambitions soared, Kai's journey into the world of competitive programming was about to intersect with a new challenge."

*[A short distance away, a group of students surrounds a figure who commands attention—Alex, a transfer student known for their expertise as a Codeforces international master. With a confident demeanor and a knack for solving complex problems, Alex effortlessly explains intricate algorithms, capturing the admiration of those around them.]*

**Yuuki:** (whispering to Kai) "That's Alex. I heard they're the go-to person for anything programming-related around here."

*[Kai glances over, observing Alex's confident smile as they effortlessly dissect the problem that had stumped everyone else in the group.]*

**Kai:** (thinking to himself) "Impressive... They're quick on their feet."

*[As Alex finishes explaining, their gaze meets Kai's briefly, a confident nod exchanged before turning back to the group.]*

**Narrator:** (voiceover) "In that moment, amidst the backdrop of eager students and spirited discussions, Kai senses a challenge—an unspoken invitation to prove himself against a formidable rival."

*[The scene transitions, setting the stage for a burgeoning rivalry that would shape Kai's journey into the competitive programming world.]*

**Scene 2**

**Scene 2: The Challenge Unveiled**

*[The university campus is alive with the buzz of excitement as the annual programming competition approaches. Banners flutter in the breeze, and students gather in groups, discussing strategies and previous contests. In a lecture hall, the Coding Club is holding a special meeting to announce the details of this year's challenge.]*

**President of the Coding Club:** "Welcome, everyone, to this year's Annual Coding Challenge! As you all know, this competition is our biggest event of the year, and it’s a chance for the best programmers to showcase their skills."

*[The audience murmurs with anticipation. Kai sits in the middle row, Yuuki beside him, both eager to hear the details, In the front row, Alex sits with a calm, confident demeanor, exuding an air of superiority.]*

**President of the Coding Club:** "This year, we have something special planned. Not only will the winner earn the prestigious title of University Champion, but they will also receive an invitation to an exclusive international competition."

*[The room buzzes with excitement at the announcement. Kai and Yuuki exchange determined glances. Alex grins to himself, knowing what's in store]*

**President of the Coding Club:** "And now, for the main event. To make things even more interesting, we will have a special duel as part of the competition. Two contestants will go head-to-head on a problem that will be revealed only at the start of their match."

*[The lights dim, and a spotlight focuses on the stage. The president gestures dramatically towards the back of the hall.]*

**President of the Coding Club:** "We have selected two top contenders for this duel: Kai and Alex!"

*[The audience erupts in applause and whispers. Kai's eyes widen in surprise, while Alex smirks confidently.]*

**Yuuki:** (whispering to Kai) "This is your chance, Kai. Show everyone what you’re made of."

**Alex:** (smirking, as he stands up) "This should be interesting. I’ve been waiting for a real challenge."

*[Kai stands up, feeling the weight of the moment but also a surge of determination.]*

**Kai:** "I won’t back down. Let’s do this."

*[The president raises their hands to quiet the room.]*

**President of the Coding Club:** "The duel will take place tomorrow at noon in the main auditorium. Be there to witness this epic showdown!"

**Narrator:** (voiceover) "The stage was set. With the eyes of their peers upon them, Kai and Alex prepared for a duel that would test their limits and define their paths in the competitive programming world."

**Scene 3**

**Scene 3: The Duel — Part 1**

*[The contest hall is alive with the hum of computers and the click-clack of keyboards. Kai and his opponent sit side by side, their faces illuminated by the glow of their screens, each tackling the challenging problem set before them: counting bracket sequences with specific constraints.]*

**Announcer:** (over intercom) "The task is simple. You have $$$n$$$ opening brackets, and $$$m$$$ closing brackets. How many of those have their longest balanced subsequence's length, equal to exactly $$$2k$$$. Your constraints are, $$$1 \le n, m, k \le 2000$$$, and $$$1 \le t \le 2000$$$ tests. Contestants, you may now begin. Good luck!"

*[The screen shows $$$n = 3, m = 2, k = 1$$$. Alongside the numbers, we had the strings ")((()", ")(()(", ")()((", "())((". Each have a longest balanced subsequence of length exactly $$$1$$$. Kai starts brainstorming while his opponent, with a confident grin, dives straight into coding.]

Alex: (muttering) "It's obvious to see that the string can be equivalently written as " (cutaway to inside his mind).

*[Let $$$c$$$ represent some number of closing brackets, $$$b$$$ represent a balanced sequence, and $$$o$$$ represent some number of opening brackets. Then we can write our problem as $$$(c + b)^t + c + o$$$ for some $$$t$$$, where the total length of $$$b$$$ is $$$k$$$. The rest of the brackets are unpaired. Addition is string concatenation, and $$$(c + b)^t$$$ is $$$(c + b)$$$ $$$t$$$ times. We just have to split the closing brackets among the $$$c$$$. This is only $$$O(n^3)$$$ per test still.]*

*[Kai struggles to find a starting point, scanning the problem statement repeatedly for clues.]*

**Kai:** (thinking) "Where do I even begin?"

*[Meanwhile, Kai's opponent progresses swiftly, reaching $$$n^3 + tn$$$ from $$$O(tn^3)$$$ ]*

Alex: (smirking) "All I need is the number of ways to combine $$$t$$$ balanced sequences, with a total length of $$$k$$$."

Kai: (internally, focusing) "Okay, $$$f(n, m, k)$$$ When we start with a closing bracket, it's like starting with $$$f(n, m - 1, k)$$$ because the first closing bracket is never matched initially. And when we start with an opening bracket, it's akin to $$$f(n - 1, m, k - 1)$$$ because it's always matched. Aha... that's our DP!"

[Kai's eyes light up with realization as the pieces of the dynamic programming recurrence fall into place.]

Kai: (thinking) "But $$$O(n^3)$$$ won't cut it... I need something faster."

[Meanwhile, Alex, confident in his approach, recognizes that his strategy can be optimized using the exponent of the Catalan generating function, a task well-suited for FFT.]

Alex: (confidently) "All I need is the exponent of the Catalan generating function. FFT will handle this effortlessly. A rookie like Kai would never think of this." Alex: (focused, calculating) "The modulus is $$$10^9 + 7$$$, not FFT-friendly... But triple mod CRT isn't about efficiency; it's about bending the rules to make FFT work, even when the author didn't intend it."

[Alex's determination intensifies as he recalibrates his approach. His mind navigates the complexities of implementing triple modular Chinese Remainder Theorem (CRT), not for efficiency, but for its sheer audacity to utilize FFT in unconventional ways.]

*[Kai, still deep in thought, considers different angles, visualizing the problem as a complex 3D cube of states and transitions.]*

**Kai:** (inner thoughts) "Each layer represents a different k... If only I could see the path clearer..."

*[Alex finishes implementing FFT and carefully writes the triple modular CRT, taking a moment to debug and refine their solution.]*

**Alex:** (focused) "Almost there... Just a few adjustments."

**Kai:** (focused) "If only I knew the base case for when $$$n$$$ or $$$m$$$ equals $$$k$$$"

*[Kai recalls his mentor's advice: "When stuck, reconsider your approach entirely." He breaks down the DP structure in his mind, realizing the solution simplifies significantly when letting the dp represent a length of at most $2k$]*

**Kai:** (epiphany) "Wait... The answer is just $$$\binom{n+m}{m}$$$! and the dp doesn't even change!!"

**Alex:** (confidently) "Done. Time to submit and secure my victory."

*[Alex clicks the submit button, his face radiating confidence as he leans back in his chair.]*

**Scene 4**

**Scene 4: The Duel — Part 2**

**Kai:** (focused) "I have to hurry..."

*[Kai, with only an idea but no concrete solution, visualizes the problem in his mind, working through each layer of the 3D cube of states.]*

*[The large screen in the auditorium lights up, displaying the progress of Alex's submission. The entire stadium falls into a tense silence, every eye glued to the unfolding results.]*

**Narrator:** (voiceover) "As Alex's submission runs on the large screen, the tension in the stadium is palpable. Every second feels like an eternity."

*[The screen displays "TLE ON TEST 1" in bold letters. Gasps and murmurs ripple through the crowd as everyone realizes that Alex's precomputation was too slow.]*

**Narrator:** (voiceover) "Time Limit Exceeded on Test 1. Alex's precomputation was too slow, and the stadium erupts in shock."

*[Alex's face flushes with embarrassment as the words "TLE ON TEST 1" flash on the screen. Despite his confidence, optimizing the FFT is no easy feat, and even Alex is momentarily stuck, re-evaluating his approach.]*

**Alex:** (muttering to himself) "How could this happen? Optimizing FFT isn't supposed to be this tricky..."

*[The audience murmurs with a mix of shock and curiosity as they watch Alex struggle to find a solution. Meanwhile, Kai sees an opportunity amidst the chaos.]*

*[Kai takes a deep breath, refocusing on his own screen. He starts to trace through his DP, looking for patterns by evaluating cases from $$$k = 0$$$ to $$$\min(n, m)$$$. His mind races, visualizing each step in a desperate bid to find a breakthrough.]*

**Kai:** (thinking to himself) "Okay, let's see... If $$$k = 0$$$ the answer is 1. What about higher values of $$$k$$$?"

*[He begins to scribble notes, mapping out the values and transitions between states, hoping to catch a glimpse of the elusive pattern.]*

**Kai:** (making calculations) "For $$$ k = 1 $$$, we have $$$ f(n, m, 1) = f(n, m - 1, 1) + f(n - 1, m, 0) $$$. But $$$ f(n - 1, m, 0) = 1 $$$, so $$$ f(n, m, 1) = f(n, m - 1, 1) + 1 $$$."

*[He verifies his calculations, scribbling down the results as he goes along.]*

**Kai:** (continuing) "And $$$ f(n, 1, 1) = n + 1 $$$, so $$$ f(n, m, 1) = n + m $$$."

*[He nods to himself, satisfied with the clarity of the derivation for $$$ k = 1 $$$, realizing it's a straightforward sum of $$$ n $$$ and $$$ m $$$.]*

**Kai:** (mentally summarizing) "So, $$$ f(n, m, 1) = n + m $$$... There's a simplicity here I've been missing."

**Alex:** (deep in thought) "All I need is the FFT of $$$ C(x) $$$, the Catalan generating function. Then, apply inverse FFT on $$$ C(x)^t $$$ for all $$$ t $$$. But inverse FFT operates independently of input values. This means... we can utilize SIMD to process all the inverse FFTs simultaneously, regardless of the specific values."

*[A sly grin spreads across Alex's face as he contemplates the power of his plan, envisioning the efficiency gained from executing multiple FFT operations in parallel, each handling different exponentiations of $$$ C(x) $$$.]*

**Alex:** (inner monologue) "Efficiency through parallelism... This will give me the edge I need."

*[With renewed determination, Alex begins to outline the meticulous steps of his strategy, confident that his advanced approach will yield swift and decisive results in the coding duel.]*

**Alex:** (confidently) "It's time to make my move."

**Scene 5**

**Scene 5: The Duel — Part 3**

**Kai:** The task is inherently symmetric, every dp value must be the same when $$$n, m$$$ are swapped. So I should be able to write every dp value using terms like $$$n + m$$$ and $$$nm$$$. But the way it looks, $$$n + m$$$ might just be enough.

**Kai:** (making calculations) "For $$$ k = 2 $$$, $$$ f(n, m, 2) = f(n, m - 1, 2) + f(n - 1, m, 1) $$$."

*[He pauses, deep in thought, recalculating the base cases and transitions in his mind.]*

**Kai:** (inner deduction) "And $$$ f(n - 1, m, 1) = n + m - 1 $$$. So, $$$ f(n, m, 2) = f(n, m - 1, 2) + (n + m - 1) $$$."

*[Kai's eyes widen as he recognizes the pattern emerging, aligning with the recursive relationship $ C(n + m, 2) = C(n + m — 1, 2) + C(n + m — 1, 1) $]*

**Kai:** (epiphany) "It's $$$ C(n + m, 2) $$$!"

*[Excitement fills Kai as he realizes he can directly compute $$$ f(n, m, 2) $$$, marking a breakthrough in his quest for an efficient solution.]*

**Kai:** (connecting the dots) "And that's it... $$$ f(n, m, k) = C(n + m, k) $$$ is the solution to my equations."

**Kai:** (inner satisfaction) "This symmetry simplifies everything."

*[With clarity now guiding his thoughts, Kai moves swiftly to implement his newfound insight, confident that he has uncovered the elegant solution to the problem.]*

**Narrator:** Alex, determined and focused, meticulously refines his SIMD implementation. Despite the setbacks and having to rewrite a significant portion of his code, he is on the verge of completing his solution.

*[In a dimly lit corner of the contest hall, Alex's screen illuminates his concentrated expression as lines of optimized code scroll rapidly. He glances at the clock, aware that every second counts in this high-stakes coding duel.]*

**Narrator:** In the heart of the contest hall, the tension was palpable as Alex and Kai raced against the clock to finalize their solutions. Both knew that the slightest misstep could cost them the victory they had been tirelessly pursuing.

*[Alex's screen flickered with lines of optimized code, his fingers moving swiftly to iron out the last bugs in his SIMD implementation. His brow furrowed with concentration as he made the final adjustments.]*

**Alex:** (focused) "Just a few more tweaks and... done! Time to submit."

*[With a click of the mouse, Alex sent his solution into the judging system, a confident smile creeping across his face. He glanced at the clock, realizing he had submitted just a fraction of a second before Kai.]*

**Narrator:** Across the room, Kai's expression was a mix of determination and relief as he put the finishing touches on his own code. His solution, rooted in a deep understanding of the problem's structure, was meticulously crafted to meet the contest's stringent requirements.

*[As Kai prepared to submit, a wave of nerves washed over him. He double-checked his code, however simple, ensuring every line was optimized and error-free. With a deep breath, he clicked the submit button, his heart pounding with anticipation.]*

*Narrator: Moments ticked by as both submissions underwent rigorous evaluation by the contest system. The giant screen above displayed progress bars inching towards completion, while the audience held their breath.*

**Announcer:** "Submission results are in progress... Evaluating..."

*[Suddenly, the screen updated. Kai's submission flashed "Compilation successful," followed by Alex's submission. But instead of a confirmation, Alex's submission showed a dreaded "Wrong Answer" on the first test case.]*

**Narrator:** Alex's heart sank as he realized the gravity of his mistake. Debug statements, meant for local troubleshooting, were still active in his code. This oversight caused his submission to output incorrect results on the very first test.

**Alex:** (frustrated) "No... debug statements! I can't believe it."

*[Swiftly, Alex retracted his submission, disabled the debug statements, and resubmitted his solution. But in the fast-paced world of competitive programming, every moment counted.]*

**Narrator:** Meanwhile, Kai's solution continued to undergo scrutiny. His careful planning and understanding of the problem's nuances paid off as his code passed test after test without a hitch.

*[As the audience held their breath, the results were finalized. Kai's solution stood firm, passing all tests flawlessly.]*

**Announcer:** "Both submissions passed! Kai and Alex submit within seconds of each other. But the delay was just enough..."

*[The room erupted into a mix of applause and sighs of sympathy. Alex's technical prowess and innovative approach were commendable, but in the unforgiving world of competitive programming, timing and precision often determined the outcome.]*

**Narrator:** Alex, known for his intense drive and unwavering determination, felt the weight of disappointment as he watched Kai's success unfold. In a moment of frustration, he slammed his hand against the table, startling those around him. The impact shattered the fragile surface, causing his laptop, provided by the competition, to crash to the ground with a resounding clatter.

*[The room fell silent as Alex, his emotions raw, realized the consequences of his outburst. He glanced at the broken table and the laptop lying on the floor, a stark reminder of his own momentary lapse.]*

**Narrator:** Without a word, Alex turned and walked away from the competition arena, his mind racing with thoughts of what could have been. The disappointment of defeat weighed heavily on him, yet somewhere deep inside, he harbored a begrudging respect for Kai's victory.