
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
Elective 4-5: Special Topics on Inequalities

Given the function $f(x)=|x-a|+|x+b|$ $(a > 0,b > 0)$.

$(1)$ If $a=1,b=2$, solve the inequality $f(x) \leqslant 5$;

$(2)$ If the minimum value of $f(x)$ is $3$, find the minimum value of $\frac{a^2}{b}+\frac{b^2}{a}$.

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