In the bustling metropolis of Econopolis, where the skyline is ever-changing and the economy thrives on savvy investments, a young entrepreneur embarks on a journey to become a real estate mogul. Gifted with a single apartment by a mysterious benefactor, this aspiring tycoon sees an opportunity to expand their empire. The strategy is simple yet ambitious: use the rental income from the apartment to acquire more, each becoming a new source of revenue. The goal? To amass a portfolio of $$$C$$$ apartments, transforming the initial gift into a sprawling empire. The journey from a single property to a real estate dynasty will require cunning, patience, and a keen eye for opportunity.

Given that you start with one apartment generating $$$B$$$ dollars per month and can purchase additional apartments for $$$A$$$ dollars each, which also start generating $$$B$$$ dollars per month once acquired, your task is to calculate how many months it will take to own $$$C$$$ apartments. Each month, you can use the income generated to buy more apartments, accelerating your journey toward becoming an apartment tycoon.

In the realm of Numeralia, a fascination with symmetry and balance pervades the culture, especially when it comes to numbers. The Numeralians hold a particular reverence for palindrome numbers, those unique numbers that read the same backward as forward, such as 323 and 111. A challenge has been set forth to the mathematicians of Numeralia: for given ranges, determine how many palindrome numbers exist between two boundaries, inclusive. This task, known as the Palindrome Count Query, tests both the mathematicians' understanding of number properties and their ability to navigate through vast numerical spaces efficiently.

You are given a task to count how many lists of $$$N$$$ positive integer numbers, each less than or equal to $$$M$$$, can be formed such that adjacent numbers in the list are relatively prime. Two numbers are relatively prime if their greatest common divisor (GCD) is 1. This problem combines number theory with combinatorial counting, challenging you to explore the intricacies of prime relationships between numbers within a constrained set.

In the bustling city of Metropoville, the demand for parking spaces is as dynamic as its ever-moving traffic. To manage this demand efficiently, the city has introduced a new parking payment system in its central parking structure, known as "Dynamic Park". Unlike traditional parking lots, Dynamic Park utilizes a tiered payment system that adjusts the cost based on the duration of parking, aiming to optimize space usage and provide fair charges for its patrons.

You've spent a day in Metropoville and now need to pay for your parking at Dynamic Park. The parking structure operates on a tiered charging system where the cost per minute changes after certain time thresholds. Given the total time you've parked, expressed in hours and minutes (for example, 19 hours and 38 minutes), calculate the total parking fee based on the tiered rates provided.

Deep within the verdant embrace of an ancient, whispering forest lies a marvel of arcane construction: the Enchanted Labyrinth. Conceived by the archmages of a forgotten era, this labyrinth guards the legendary Crystal of Aeternum, a gem of untold power. While many have braved the labyrinth's convoluted corridors in search of glory, only the echoes of their valor remain, a testament to the maze's perilous nature. The labyrinth is a network of $$$N$$$ mystic chambers, linked by $$$M$$$ pathways of uniform length, with only $$$K$$$ chambers harboring portals that lead back to the realm of men.

Elisa, a daring explorer driven by tales of the ancient relic, finds herself ensnared within the labyrinth's spellbound walls. From the initial chamber labeled $$$1$$$, she must navigate through the maze to reach a portal that promises freedom. However, the labyrinth is not without its sentinel – a cunning Minotaur, tasked with thwarting the escape of intruders by manipulating the very fabric of the maze.

The Minotaur, bound to the magic of the labyrinth, has the power to seal any one pathway leading from the chamber in which Elisa currently resides, effectively blocking her passage through that route. However, its power is limited to the chamber's confines at Elisa's location, and once she embarks upon an unsealed pathway, the beast cannot obstruct her until she arrives at the subsequent chamber. In this deadly game of wits and wills, your challenge is to determine the shortest path that Elisa can take to ensure her escape or to despairingly conclude that the Minotaur's guile renders escape impossible, in which case, return $$$-1$$$.

In the heart of the enchanting town of Graphville lies Vertex Park, an expanse of lush landscapes intertwined with a complex network of pathways. This park, renowned for its verdant groves and serene ambiance, serves as a cherished rendezvous for three inseparable friends: Alice, Bob, and Carol. Unlike the typical parks, Vertex Park boasts a unique design resembling a tree structure, with pathways ensuring direct and loop-free connections between its various junctions.

One sunny afternoon, inspired by the interconnectedness of the park, the trio conceives a playful challenge: starting from different locations within the park, they aim to ascertain the minimal steps necessary for a joint gathering at a single junction. This venture seeks not just to test their understanding of the park's layout but to maximize their time together amidst nature's bounty. Given the uniform length of the pathways and the strategic starting points of Alice, Bob, and Carol, your task is to calculate the fewest steps required for their reunion, thereby facilitating an extended companionship in their verdant haven.

In the realm of Gridtopia, adventurers find themselves facing a sprawling $$$N \times M$$$ grid. Starting from the upper left corner, their goal is to traverse to the lower right corner, collecting scattered magical artifacts along the way. However, Gridtopia has a peculiar rule: movement is restricted to only rightward or downward steps. Furthermore, each cell of the grid may contain at most one artifact. Given the adventurers are faced with the challenge of planning the minimum number of trips required to ensure all artifacts are collected and no treasure is left unclaimed in their quest across Gridtopia.

In the digital expanse of Textland, a land renowned for its affinity with strings and textual puzzles, a challenge has arisen that has piqued the curiosity of its inhabitants. The challenge involves a mystical string, said to hold secrets to the ancient algorithms of Textland. This string, composed of lowercase letters, is rumored to contain hidden patterns that repeat throughout its length. The quest is to uncover the most repeated substring within this enigmatic string. However, the challenge comes with a twist: if multiple substrings are equally repeated, the quest seeks the longest among them. Should a tie still persist, the preference shifts to the substring that is smallest lexicographically. The citizens of Textland are intrigued, for deciphering this puzzle would mean unlocking a piece of the lost lore of their digital realm.

In the mystical realm of Lexicovia, a land where magic weaves through words and letters with the power to manifest the deepest desires, a group of friends skilled in the art of Spellcraft gather. These friends, known throughout the land for their wisdom and camaraderie, possess a unique collection of enchanted words—each a spell in its own right. As the annual Festival of the Arcane Words approaches, they embark on a quest to combine their spells in a way that summons a specific magical effect, encapsulated in a secret word known only to them. This year, the word they seek to manifest is a symbol of their enduring bond, a testament to the magic of friendship that has kept them united through every trial.

Your task is to assist this circle of friends in determining whether they can arrange their collection of enchanted words in such a manner that, when combined without altering the sequence of letters within each word, they can create a magical phrase that contains a specific secret word $$$S$$$ as a substring. The magical essence of their spells allows for the words to be reordered but forbids any alteration to the letters within each word, preserving the spell's integrity.