
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
If $x^{n}=y$, then we denote it as $\left(x,y\right)=n$. For example, $3^{2}=9$, then $\left(3,9\right)=2$.  
$(1)$ According to the above rule, fill in the blanks: $\left(10,1000\right)=\_\_\_\_\_\_$, $\left(-5,25\right)=\_\_\_\_\_\_$;  
$(2)$ If $\left(x,16\right)=2$, then $x=$______;  
$(3)$ If $\left(4,a\right)=2$, $\left(b,8\right)=3$, find the value of $\left(b,a\right)$.

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