
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
Let $\{a_n\}$ be a sequence of positive terms, and its sum of the first $n$ terms $S_n$ satisfies: $4S_n = (a_n - 1)(a_n + 3)$. Find the value of $a_{1005}$.

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