
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
Let $A_{n}=\{1,2,3,\ldots ,n\}$ $(n\geqslant 2, n\in \mathbb{N})$, and let $P\subseteq A_{n}$. If for any element $x\in P$, either $x-1\in P$ or $x+1\in P$, then the set $P$ is called a self-neighbor set of $A_{n}$. Let $a_{n}(k)$ $(1\leqslant k\leqslant n, k\in \mathbb{N})$ be the number of sets of all self-neighbor sets of $A_{n}$ with the largest element $k$.  
$(1)$ Determine directly whether the sets $P=\{1,2,3,5\}$ and $Q=\{1,2,4,5\}$ are self-neighbor sets of $A_{5}$.  
$(2)$ Compare $a_{10}(6)$ and $a_{10}(5)+a_{10}(3)$, and explain the reason.

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