
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 sequence $\{a_{n}\}$ such that $a_{1}=2$, $a_{2}=4$, and the sequence $\{a_{n}-n\}$ is a geometric sequence.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Find the sum: $S_{n}=a_{1}+a_{2}+\cdots +a_{n}$.

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