
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( x_0 = x_{101} = 0 \). The numbers \( x_1, x_2, \ldots, x_{100} \) are chosen at random from the interval \([0, 1]\) uniformly and independently. Compute the probability that \( 2x_i \geq x_{i-1} + x_{i+1} \) for all \( i = 1, 2, \ldots, 100 \).

Present the answer in LaTex format: \boxed{Your answer}
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