
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 $\left\{a_n\right\}$ with a non-zero common difference, it satisfies $a_1 + a_3 + a_5 = 12$, and $a_1, a_5, a_{17}$ forms a geometric sequence. Let $S_n$ be the sum of the first $n$ terms of $\left\{a_n\right\}$.

(1) Find the general formula for the $n$-th term of the sequence $\left\{a_n\right\}$.

(2) Find the maximum positive integer $n$ such that $S_n < 5a_n$.

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