
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 infinite sequence $\{a_{n}\}$ that satisfies: if $a_{m}=a_{n}$, then $a_{m+1}=a_{n+1}$, and $a_{1}=a_{5}=1$, $a_{3}=-3$, $a_{4}=4$, and $a_{2}$ is the geometric mean of $a_{1}$ and $a_{4}$. Find the maximum value of the sum of the first $n$ terms of $\{a_{n}\}$.

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