
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 $\{a\_n\}$ where all terms are positive, it is known that $a\_4 = 2a\_2$, and $a\_1$, $4$, $a\_4$ form a geometric sequence.

(1) Find the general term formula for the sequence $\{a\_n\}$.
(2) Find the sum of all $a\_n$ that satisfy the following conditions: ① $20 ≤ n ≤ 116$; ② $n$ is divisible by $5$.

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