
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
Determine the smallest odd integer  $n \ge 3$ , for which there exist  $n$  rational numbers  $x_1 , x_2 , . . . , x_n$  with the following properties: $(a)$  \[\sum_{i=1}^{n} x_i =0 , \  \sum_{i=1}^{n} x_i^2 = 1.\] $(b)$  \[x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.\]

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