
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
Find all positive integers  $n$  such that the following statement holds: Suppose real numbers  $a_1$ ,  $a_2$ ,  $\dots$ ,  $a_n$ ,  $b_1$ ,  $b_2$ ,  $\dots$ ,  $b_n$  satisfy  $|a_k|+|b_k|=1$  for all  $k=1,\dots,n$ . Then there exists  $\varepsilon_1$ ,  $\varepsilon_2$ ,  $\dots$ ,  $\varepsilon_n$ , each of which is either  $-1$  or  $1$ , such that
    \[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]

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