
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  $n>1$  be an integer. For each numbers  $(x_1, x_2,\dots, x_n)$  with  $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$ , denote $m=\min\{|x_i-x_j|, 0<i<j<n+1\}$ Find the maximum value of  $m$ .

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