
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
For a finite sequence $\{a_{n}\}$, $n\leqslant N$, $N\geqslant 3$, $N\in N^{*}$, define: for any $k\leqslant N$, $k\in N^{*}$, we have:<br/>$(1)S^{*}(k)=|a_{1}|+|a_{2}|+|a_{3}|+\cdots +|a_{k}|$;<br/>$(2)$ for $c\in R$, let $L\left(k\right)=|a_{1}-c|+|a_{2}-c|+|a_{3}-c|+\cdots +|a_{k}-c|$.<br/>For $k\in N^{*}$, if there exists a non-zero constant $c$ such that $L\left(k\right)=S^{*}(k)$, then the constant $c$ is called the $k$-th $\omega$ coefficient of the sequence $\{a_{n}\}$.<br/>$(Ⅰ)$ Suppose the general term of the sequence $\{a_{n}\}$ is ${a_n}={(-2)^n}$, calculate $S^{*}(4)$, and determine if $2$ is the $4$-th $\omega$ coefficient of the sequence;<br/>$(Ⅱ)$ Suppose the general term of the sequence $\{a_{n}\}$ is $a_{n}=3n-39$, and the $m$-th $\omega$ coefficient of the sequence $\{a_{n}\}$ is $3$, find the value of $m$;<br/>$(Ⅲ)$ Suppose the sequence $\{a_{n}\}$ is an arithmetic sequence, and both $-1$ and $2$ are the $m$-th $\omega$ coefficients of the sequence $\{a_{n}\}$, and $S^{*}(m)=507$, find the maximum value of $m$.

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