It was a digit DP problem. The states of the dp could be dp[index][previous][size - of - current - sequence][constraint][increasing / decreasing][mask].
Index — This shows the length of the number formed.
previous — This is used to store the last digit used.
size - of - current - sequence — This shows the size of the current sequence. It has to be less than K.
constraint — This shows whether the current number formed is constrained or not.
increasing / decreasing — This is used to know whether we are currently making an increasing or decreasing sequence.
mask — This is used to take care of the condition that given M digits occur at least once in the number.

This is a geometry problem which can be solved using rotating line sweep algorithm. We iterate over all the lines(targets) and for each end of that line we will calculate and store the angle of all the other endpoints of the (n - 1) other targets.
And we will sort these values according to the angle. Then we will rotate the line passing through P by iterating over the other points, in slope order (We have already sorted it in slope order). If we encounter the first end of some target then we will add the length of that target to our current point and as we reach the other end we will subtract the length of that line from our current point. We have to perform the same for all the n given targets. We have to perform the same and take the maximum also we have add 1 when we reach the first end of some target to count the number of targets and as we reach the other end subtract 1 and in this way we can count the number of targets required to be destroyed to get current points.

The time complexity would be O(n * n * log(n)). The first n comes from iterating over all the n targets and the n * log(n) factor comes from storing the angle and sorting it.

It was based on Matrix Exponentiation. The initial matrix to be formed will be a 64 * 64 matrix, where we store the number of ways to go from a 4 letter state to another 4 letter state, keeping in mind all the restrictions.
However, this matrix with 10⁴ test cases will lead to TLE, with total operations being 10⁴ * (64³log n).
Hence, with the solutions to initial values, one can solve the linear equations using any method say Gauss Elimination, and obtain a much smaller 3 * 3 matrix, thereby reducing the number of operations to 10⁴ * (3³log n)