
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  $n$  be a given integer such that  $n\ge 2$ . Find the smallest real number  $\lambda$  with the following property: for any real numbers  $x_1,x_2,\ldots ,x_n\in [0,1]$  , there exists integers  $\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\}$  such that the inequality   $$ \left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambda $$  holds for all pairs of integers  $(i,j)$  where  $1\le i\le j\le n$ .

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