
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 that $\{a_{n}\}$ is an arithmetic sequence, $a_{1}=1$, $d\neq 0$, and $a_{1}$, $a_{2}$, $a_{4}$ form a geometric sequence.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Let $b_{n}=a_{n(n+1)}$, and define ${S}_{n}=-{b}_{1}+{b}_{2}-{b}_{3}+\cdots+{(-1)}^{n}{b}_{n}$. Find $S_{n}$.

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