
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 $f\left(x\right)=|2x-3|+|x-2|$.<br/>$(1)$ Find the solution set $M$ of the inequality $f\left(x\right)\leqslant 3$;<br/>$(2)$ Under the condition of (1), let the smallest number in $M$ be $m$, and let positive numbers $a$ and $b$ satisfy $a+b=3m$, find the minimum value of $\frac{{{b^2}+5}}{a}+\frac{{{a^2}}}{b}$.

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