
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: $$\frac{\sin^{2}a_{3} - \cos^{2}a_{3} + \cos^{2}a_{3}\cos^{2}a_{6} - \sin^{2}a_{3}\sin^{2}a_{6}}{\sin(a_{4} + a_{5})} = 1$$, with common difference $d \in (-1, 0)$. If and only if $n=9$, the sum of the first $n$ terms of the sequence, denoted as $S_n$, reaches its maximum value. Determine the range of the first term $a_1$.

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