
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 $f(x)=|\frac{1+2x}{x}|+|\frac{1-2x}{x}|$ has a minimum value of $m$.  
$(1)$ Find the value of $m$;  
$(2)$ If $a \gt 0$, $b \gt 0$, and $a+b=m$, find the maximum value of $\frac{ab}{a+4b}$.

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