
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
In general, $\frac{a}{2}+\frac{b}{3}=\frac{a+b}{2+3}$ is not true, but there are some numbers that can make it true, for example: $a=b=0$. We call a pair of numbers $a$ and $b$ that make $\frac{a}{2}+\frac{b}{3}=\frac{a+b}{2+3}$ true as "companion number pairs," denoted as $\left(a,b\right)$.  
$(1)$ If $\left(a,1\right)$ is a "companion number pair," find the value of $a$;  
$(2)$ Write down a "companion number pair" $\left(a,b\right)$ where $b\neq 0$ and $b\neq 1$;  
$(3)$ If $\left(m,n\right)$ is a "companion number pair," find the value of the algebraic expression $14m-5n-\left[5m-3\left(3n-1\right)\right]+3$.

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