
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 the sequence {a<sub>n</sub>}, the sum of its first n terms is S<sub>n</sub>, and a<sub>n</sub> + a<sub>n+2</sub> = 2a<sub>n+1</sub>. Also, a<sub>1</sub> = 13 and a<sub>2</sub> = 11. Determine the value of n when S<sub>n</sub> reaches its maximum value.

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