
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 $\{a_{n}\}$, with ${a}_{1}=1,{a}_{n+1}=2{a}_{n}-cos\frac{nπ}{2}+2sin\frac{nπ}{2}(n∈{N}^{*})$.<br/>$(1)$ Find $a_{2}$ and $a_{3}$;<br/>$(2)$ Find the general formula for $\{a_{n}\}$;<br/>$(3)$ Let the sum of the first $n$ terms of the sequence $\{n({a}_{n}-{2}^{n})\}$ be $T_{n}$, if ${T}_{m}=2024(m∈{N}^{*})$, find $m$.

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