
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
Let the function $f(x)=|x-a|$ where $(a > 0)$.  
$(1)$ When $a=2$, solve the inequality $f(x) \geqslant 1-2x$;  
$(2)$ Given that the minimum value of $f(x)+|x-1|$ is $3$, and $m^{2}n=a$ where $(m > 0,n > 0)$, find the minimum value of $m+n$.

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