
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 $\{a\_n\}$ where all terms are positive, the sum of the first $n$ terms is denoted as $S\_n$, and it satisfies $S\_n = 4 - a\_n$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $b\_n$ be defined as follows: $b\_n = \begin{cases} \log_{\frac{1}{2}} a\_n & \text{if $n$ is odd} \\ a\_n & \text{if $n$ is even} \end{cases}$ for all $n \in \mathbb{N}^*$. Find the sum of the first $2n$ terms of the sequence $\{b\_n\}$, denoted as $T_{2n}$.

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