
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 each term is positive and it satisfies: $a_4 = 2a_2$, and $a_1$, 4, $a_4$ form a geometric sequence.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Find the sum of all $a_n$ that satisfy the following conditions: ① $20 \leq n \leq 116$; ② $n$ is divisible by 5.

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