
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
Denote by  $\mathbb Z^2$  the set of all points  $(x,y)$  in the plane with integer coordinates.  For each integer  $n\geq 0$ , let  $P_n$  be the subset of  $\mathbb Z^2$  consisting of the point  $(0,0)$  together with all points  $(x,y)$  such that  $x^2+y^2=2^k$  for some integer  $k\leq n$ .  Determine, as a function of  $n$ , the number of four-point subsets of  $P_n$  whose elements are the vertices of a square.

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