
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
The sequence $\{a_n\}$ consists of positive terms, and $S_n$ denotes the sum of its first $n$ terms. For any $n \in \mathbb{N}^*$, it always holds that $a_n$, $S_n$, $a_n^2$ form an arithmetic sequence. Determine the value of $a_{2009}$.

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