
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 non-zero common difference arithmetic sequence $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$, it satisfies $S\_3 = a\_4 + 4$. Additionally, $a\_2$, $a\_6$, and $a\_18$ form a geometric sequence.

(1) Find the general term formula for the sequence $\{a\_n\}$.

(2) Let $b\_n = \frac{a\_n}{2^n}$, find the sum of the first $n$ terms for the sequence $\{b\_n\}$, denoted as $T\_n$.

(3) Let $c\_n = \sqrt{S\_n + t}$, if $\{c\_n\}$ is an arithmetic sequence, find the value of the real number $t$.

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