
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 different monic quadratic trinomials (i.e. with the leading coefficient equal to 1) with integer coefficients that have two different roots, which are powers of the number 3 with natural exponents, and their coefficients in absolute value do not exceed $27^{45}$.

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