
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
Determine the number of positive integers $n$ for which the logarithms of the sides of a triangle can be $\log_{10} (2 \cdot 12)$, $\log_{10} (2 \cdot 75)$, and $\log_{10} \sqrt{n}$. The triangle is nondegenerate and has a positive area.

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