
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 the number of distinct monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have at least one root, all of their roots are powers of 5 with non-negative integer exponents, and all of their coefficients in absolute value do not exceed $122^{85}$.

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