
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
Given a positive integer \( n \), find the smallest real number \( \lambda \) such that there exist real numbers \( a_1, a_2, \ldots, a_n \) within the interval \([0, 1]\), for any real numbers \( x_1, x_2, \ldots, x_n \) satisfying \( 0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq 1 \), we have
$$
\min_{1 \leq i \leq n} \left|x_i - a_i\right| \leq \lambda
$$

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