
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
Given the following definition: We call a pair of rational numbers $a$ and $b$ that satisfy the equation $a-b=ab+1$ as "companion rational number pairs," denoted as $\left(a,b\right)$. For example, $3-\frac{1}{2}=3\times \frac{1}{2}+1$, $5-\frac{2}{3}=5\times \frac{2}{3}+1$, so the pairs $(3,\frac{1}{2})$ and $(5,\frac{2}{3})$ are both "companion rational number pairs."

$(1)$ Among the pairs $(-2,\frac{1}{3})$ and $(-\frac{1}{2},-3)$, the "companion rational number pair" is ______;

$(2)$ If $\left(x+1,5\right)$ is a "companion rational number pair," then the value of $x$ is ______;

$(3)$ If $\left(a,b\right)$ is a "companion rational number pair," find the value of $3ab-a+\frac{1}{2}(a+b-5ab)+1$.

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