Consider the following problem: You are given a list of integers. Write a Python function that finds the longest increasing subsequence in the list. A subsequence is considered increasing if the elements are in ascending order, but they may not necessarily be contiguous. Your function should return the length of the longest increasing subsequence. For example, given the list [3, 4, 2, 8, 10, 5, 1], the longest increasing subsequence is [3, 4, 8, 10], so the function should return 4. Provide the implementation of the function step by step.

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