
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
Suppose that  $F$  is a field with exactly  $5^{14}$  elements. We say that a function  $f:F \rightarrow F$  is *happy*, if, for all  $x,y \in F$ ,   $$ \left(f(x+y)+f(x)\right)\left(f(x-y)+f(x)\right)=f(y^2)-f(x^2). $$  Compute the number of elements  $z$  of  $F$  such that there exist distinct happy functions  $h_1$  and  $h_2$  such that  $h_1(z)=h_2(z).$ *Proposed by Luke Robitaille*

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