
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 a polynomial $P(x, y)$ with integer coefficients such that:

1. The number of integers that $P(x, y)$ attains when $P(x, y)>0$ and $x, y$ are positive integers is infinite.
2. If $x, y$ are positive integers such that $P(x, y)$ is a strictly positive integer, then $P(x, y)$ is a Fibonacci number. Recall that the Fibonacci sequence is defined by $F_0=1, F_1=1$ and by the recurrence relation $F_{n+1}=F_n+F_{n-1}$ for $n \geq 1$. The integers $F_0, F_1, F_2, \ldots$ are called Fibonacci numbers.

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