
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
Suppose that \( a_{1}, a_{2}, a_{3}, a_{4}, \cdots \) is an arithmetic progression with \( a_{1} > 0 \) and \( 3 a_{8} = 5 a_{13} \). Let \( S_{n} = a_{1} + a_{2} + \cdots + a_{n} \) for all integers \( n \geq 1 \). Find the integer \( n \) such that \( S_{n} \) has the maximum value.

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