
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 $f\left(x\right)=|x+1|+|1-2x|$.<br/>$(1)$ Find the solution set for the inequality $f\left(x\right)\leqslant 3$;<br/>$(2)$ If $0 \lt b \lt \frac{1}{2} \lt a$, and $f\left(a\right)=3f\left(b\right)$, $a^{2}+b^{2} \gt m\left(m\in Z\right)$ always holds, find the maximum value of $m$.

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