
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 $f(x)=x^{2}-2ax+b$ where $a, b \in \mathbb{R}$.  
$(1)$ If the solution set of the inequality $f(x) \leqslant 0$ is $\left[-1,2\right]$, find the solution set of the inequality $bx^{2}-2ax+1\leqslant 0$;  
$(2)$ If $b=a^{2}$, and for any $x_{1}\in \left[2,4\right]$, there exists $x_{2}\in \left[2,4\right]$ such that $f(x_{1})f(x_{2})=1$, find the value of the real number $a$.

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