
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
In the geometric sequence $\{a_n\}$, it is given that $a_{13}=1$ and $a_{12} > a_{13}$. Find the largest integer $n$ for which $(a_1-\frac{1}{a_1})+(a_2-\frac{1}{a_2})+(a_3-\frac{1}{a_3})+\cdots+(a_n-\frac{1}{a_n}) > 0$.

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