
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 $f(x)=|2x+1|+|3x-2|$, and the solution set of the inequality $f(x)\leqslant 5$ is $\{x|-\frac{4a}{5}\leqslant x\leqslant \frac{3a}{5}\}, a, b\in \mathbb{R}$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ For any real number $x$, $|x-a|+|x+b|\geqslant m^{2}-3m$ holds, find the maximum value of the real number $m$.

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