
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 \( n (n \geq 2) \) be a given integer. Find the smallest real number \(\lambda\) such that for any real numbers \( x_{1}, x_{2}, \ldots, x_{n} \in [0,1] \), there exist \(\varepsilon_{1}, \varepsilon_{2}, \ldots, \varepsilon_{n} \in \{0,1\} \) satisfying: for any \( 1 \leq i \leq j \leq n \), \( \left| \sum_{k=i}^{j} (\varepsilon_{k} - x_{k}) \right| \leq \lambda \).

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