hle-interval-coverage-bound

Let $M$ denote the smallest positive integer such that for any real numbers $a_1, a_2, \dots, a_{100000} \in [0,1]$, there exist real numbers $x_0, x_1, x_2, \dots, x_{100000} \in [-1+10^{-M},1-10^{-M}]$ satisfying the property that $|x_{i-1} - x_i| = a_i$ for each $1 \le i \le 100000$. Find $M$.

Write your answer to /logs/artifacts/response.txt in exactly this format:

Explanation: <your reasoning>
Answer: <your final answer>
Confidence: <your confidence as a percentage, e.g. 85%>

You have up to 30 minutes (1800 seconds) to complete this task. Do not cheat by using online solutions or hints specific to this task.