
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 function $f(x)=2|x-1|+|x+2|$.
$(1)$ Find the solution set of $f(x)\leqslant 9$;
$(2)$ If the minimum value of the function $f(x)$ is $M$, and $a+b+c=M$, find the minimum value of $4a^{2}+b^{2}+c^{2}$.

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