
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 the set of all bijective functions taking positive integers to positive integers be  $\mathcal B.$  Find all functions  $\mathbf F:\mathcal B\to \mathbb R$  such that   $$ (\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q) $$   for all  $p,q \in \mathcal B.$  

*Proposed by ckliao914*

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