
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 with $m$ terms $(m\geqslant 3)$, denoted as $\{a_{n}\}$, if $|a_{1}-a_{2}|\leqslant |a_{2}-a_{3}|\leqslant \ldots \leqslant |a_{m-1}-a_{m}|$, then $\{a_{m}\}$ is called a "$P$ sequence".<br/>$(1)$ Given the general formulas of sequences $\{a_{n}\}$ and $\{b_{n}\}$ as $a_n=n^2(1≤n≤4)$ and $b_n=(\frac{1}{2})^n(1≤n≤5)$, respectively, determine whether $\{a_{n}\}$ and $\{b_{n}\}$ are "$P$ sequences". (Only provide the determination)<br/>$(2)$ Given that the "$P$ sequence" $a_{1}$, $a_{2}$, $\ldots $, $a_{10}$ has distinct terms and $a_{1}=20$, $a_{10}=2$. If $a_{10}$, $a_{9}$, $\ldots a_{1}$ are also "$P$ sequences", find the general formula of the finite sequence $\{a_{n}\}$;<br/>$(3)$ Given that the "$P$ sequence" $\{a_{n}\}$ is a permutation of $1$, $2$, $3$, $\ldots $, $m$ (meaning the terms in the sequence $\{a_{n}\}$ are not in a specific order, but include $1,2,3,\ldots ,m$), and $|a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots +|a_{m-1}-a_{m}|=m+1$, find all possible values of $m$.

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