
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
Observe the following two equations: $2-\frac{1}{3}=2\times \frac{1}{3}+1$, $5-\frac{2}{3}=5\times \frac{2}{3}+1$. We define as follows: we call a pair of rational numbers $a$ and $b$ that satisfy the equation $a-b=ab+1$ as "coexistent rational number pairs," denoted as $\left(a,b\right)$. For example, the pairs $(2,\frac{1}{3})$ and $(5,\frac{2}{3})$ are both "coexistent rational number pairs".
$(1)$ The "coexistent rational number pair" among the pairs $\left(-2,1\right)$ and $(3,\frac{1}{2})$ is ______;
$(2)$ If $\left(m,n\right)$ is a "coexistent rational number pair," then $\left(-n,-m\right)$ is ______ a "coexistent rational number pair" (fill in "is" or "is not");
$(3)$ Please provide another pair of "coexistent rational number pairs" that satisfy the conditions as ______ (Note: cannot repeat the "coexistent rational number pairs" already mentioned in the problem);
$(4)$ If $\left(a,3\right)$ is a "coexistent rational number pair," find the value of $a$.

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