
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
Define the $R_{p}$ sequence $\{a_{n}\}$: for $p\in R$, satisfying:<br/>①$a_{1}+p\geqslant 0$,$a_{2}+p=0$; ②$\forall n\in N*$, $a_{4n-1} \lt a_{4n}$; ③$\forall m$, $n\in N*$, $a_{m+n}\in \{a_{m}+a_{n}+p$, $a_{m}+a_{n}+p+1\}$.<br/>$(1)$ For the sequence of the first $4$ terms $2$, $-2$, $0$, $1$, can it be an $R_{2}$ sequence? Explain the reason; <br/>$(2)$ If $\{a_{n}\}$ is an $R_{0}$ sequence, find the value of $a_{5}$; <br/>$(3)$ If $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, does there exist $p\in R$ such that there exists an $R_{p}$ sequence $\{a_{n}\}$, for any $n\in N*$, satisfying $S_{n}\geqslant S_{10}$? If it exists, find all such $p$; if it does not exist, explain the reason.

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