
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 $f(x)=2|x|+|2x-m|$ where $m>0$, and the graph of the function is symmetric about the line $x=1$.  
$(Ⅰ)$ Find the minimum value of $f(x)$.  
$(Ⅱ)$ Let $a$ and $b$ be positive numbers such that $a+b=m$. Find the minimum value of $\frac{1}{a}+\frac{4}{b}$.

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