
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 its first $n$ terms $S_n$, it is known that $S_1+1$, $S_3$, $S_4$ form an arithmetic sequence, and $a_1$, $a_2$, $a_5$ form a geometric sequence.

1. Find the general term formula for the sequence ${a_n}$;
2. If $S_4$, $S_6$, $S_n$ form a geometric sequence, find $n$ and the common ratio of this geometric sequence.

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