
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]


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Given that the set $M$ consists of all functions $f(x)$ that satisfy the following property: there exist real numbers $a$ and $k(k \neq 0)$, such that for any $x$ in the domain, $f(a+x)=kf(a-x)$ holds. The pair $(a,k)$ is called the "accompanying pair" of the function $f(x)$.

$(1)$ Determine whether $f(x)=x^{2}$ belongs to set $M$, and explain the reason;

$(2)$ If the function $f(x)=\sin x \in M$, find all the "accompanying pairs" of the function $f(x)$ that satisfy the condition;

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