
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 an arithmetic sequence $\{a\_n\}$ with the first term of 1, the sum of the first $n$ terms is $S\_n$. It is known that $S\_3$ is the arithmetic mean of $a\_4$ and $a\_5$. Another sequence $\{b\_n\}$ satisfies $b\_n = 2^{\frac{S\_n + n}{2n}}$.

1. Find the general terms of the sequences $\{a\_n\}$ and $\{b\_n\}$.
2. Find the sum of the first $n$ terms, denoted as $T\_n$, of the sequence $\{a\_n \cdot b\_n\}$.

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