
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
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $a_8 > 0$, $a_8 + a_9 < 0$. The maximum value of $n$ for which $S_n > 0$ is \_\_\_\_\_\_; the largest term in the sequence $\left\{\frac{S_n}{a_n}\right\}$ (for $1 \leq n \leq 15$) is the \_\_\_\_\_\_ term.

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