
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 an arithmetic sequence, $\{b\_n\}$ is a geometric sequence, $T\_n$ is the sum of the first $n$ terms of $\{b\_n\}$, $a\_1=b\_1=1$, and $\sqrt {a\_2+2}+ \sqrt {b\_2-2}=2 \sqrt {2}$, find the minimum value of $a\_2+b\_2$. Then,
(1) Find $T\_n$;
(2) Let $S\_n$ be the sum of the first $n$ terms of $\{|a\_n|\}$. Find $S\_n$.

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