
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 the sequence $\{a_{n}\}$ is an arithmetic sequence, ${b}_{n}=\left\{\begin{array}{l}{a}_{n}+5,n=2k-1,k∈{N}^{*}\\-{a}_{n}+10,n=2k,k∈{N}^{*}\end{array}\right.$, let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$, and $T_{3}=T_{4}=24$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If $T_{n}=T_{m}=96\left(n \lt m\right)$, find $n$ and $m$.

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