
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 $f(x)=|x-a|$.
$(1)$ If $a=1$, solve the inequality $f(x) \geqslant 4-|x+1|$;
$(2)$ If the solution set of the inequality $f(x) \leqslant 1$ is $[0,2]$, and $\frac {1}{m}+ \frac {1}{2n}=a \ (m > 0,n > 0)$, find the minimum value of $mn$.

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