
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
Given an arithmetic sequence $\{a_n\}$ that satisfies: $3a_7=5a_{13}$, and $\cos^2a_4-\cos^2a_4\sin^2a_7+\sin^2a_4\cos^2a_7-\sin^2a_4=-\cos(a_5+a_6)$. The common difference $d$ is in the interval $(2,0)$. Find the maximum value of the sum of the first $n$ terms of the sequence $\{a_n\}$.

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