
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 $\{a_n\}$ is a geometric sequence with the sum of the first $n$ terms denoted as $S_n (n\in\mathbb{N^*})$, and $\frac{1}{a_1}-\frac{1}{a_2}=\frac{2}{a_3}$, $(S_6=63)$.

(1) Find the general term formula for $\{a_n\}$;
(2) If for any $n\in\mathbb{N^*}$, $b_n$ is the arithmetic mean of $\log_2{a_n}$ and $\log_2{a_{n+1}}$, find the sum of the first $2n$ terms of the sequence $\{(−1)^n b_n^2\}$.

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