
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 two sequences $\{a\_n\}$ and $\{b\_n\}$ where all terms are non-zero, $\{b\_n\}$ is a monotonically increasing sequence, and $2a\_n = b\_n \cdot b\_{n+1}$, $a\_n + a\_{n+1} = b\_{n+1}^2$, $a\_1 = b\_2$, and $a\_2 = b\_6$.

1. Find $b\_1$ and the general term formula for the sequence $\{b\_n\}$.
2. If sequence $\{c\_n\}$ satisfies $c\_1 = -\frac{1}{3}$ and $c\_n + c\_{n+1} = (\sqrt{2})^{b\_n}$, find the sum $S\_n$ of the first $n$ terms of the sequence $\{c\_{2n}\}$.

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