
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 \( n \geqslant 3 \) such that for any real numbers \( a_{1}, a_{2}, \ldots, a_{n} \) and \( b_{1}, b_{2}, \ldots, b_{n} \), if \( \left|a_{k}\right| + \left|b_{k}\right| = 1 \) for each integer \( k \) (where \( 1 \leqslant k \leqslant n \)), there exist \( x_{1}, x_{2}, \ldots, x_{n} \in \{-1,1\} \) such that
\[ 
\left|\sum_{k=1}^{n} x_{k} a_{k}\right| + \left|\sum_{k=1}^{n} x_{k} b_{k}\right| \leqslant 1 .
\]

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