
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  $n$ -tuples of real numbers  $(x_1, x_2, \dots, x_n)$  such that, for every index  $k$  with  $1\leq k\leq n$ , the following holds:

\[ x_k^2=\sum\limits_{\substack{i < j  i, j\neq k}} x_ix_j \]

*Proposed by Oriol Solé*

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