
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 a sequence $\{a_n\}$ and its sum of the first $n$ terms $S_n$, satisfying $a_1=2$, $a_{n+1}=2S_n+2$ $(n \geqslant 1)$,

$(1)$ Find the general formula for $\{a_n\}$;

$(2)$ If the sequence $\{b_n\}$ satisfies: $b_n=a_n+(-1)^n\log_3a_n$, find the sum of the first $2n$ terms of $\{b_n\}$, $S_{2n}$.

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