
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 that all terms are positive in the geometric sequence $\{a_n\}$, and $-a_3, a_2, a_4$ form an arithmetic sequence,

$(1)$ If $a_1=1$, find the sum of the first $n$ terms of $\{a_n\}$, denoted as $S_n$;

$(2)$ If $b_n=\log_2 a_{2n+1}$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n=n^2+3n$, find $a_1$.

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