
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 $\{a\_n\}$ with a common difference $d∈(0,1)$, and $\frac {\sin ^{2}a_{3}-\sin ^{2}a_{7}}{\sin (a_{3}+a_{7})}=-1$, if $a_{1}∈(-\frac {5π}{4},-\frac {9π}{8})$ then the minimum value of $n$ for which the sum of the first $n$ terms of the sequence, $S_{n}$, is minimized is $\_\_\_\_\_\_$.

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