
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
Let $\{a_{n}\}$ be an arithmetic sequence with the first term $1$ and a common difference not equal to $0$, and $a_{1}$, $a_{2}$, $a_{6}$ form a geometric sequence.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let $b_{n} = (-1)^{n}a_{n}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted by $S_{n}$.

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