
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 $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\{b_{n}\}$ is an arithmetic sequence with a common difference of $-1$.  
$(1)$ Investigation: Determine whether the sequence $\{a_{n}-n\}$ is an arithmetic sequence or a geometric sequence, and provide a justification.  
$(2)$ Find the smallest positive integer value of $n$ that satisfies $a_{1}+a_{2}+\ldots +a_{n} \gt 2200$.

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