
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
In an arithmetic sequence $\{a_{n}\}$, where $a_{1}=25$, $a_{2}\neq a_{1}$, and $a_{1}$, $a_{11}$, $a_{13}$ form a geometric sequence. The sum of the first $n$ terms of $\{a_{n}\}$ is denoted as $S_{n}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Determine the maximum value of $S_{n}$.

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