
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 $f\left(x\right)=|2x-1|+|2x-k|$.<br/>$(1)$ If $f\left(x\right)\geqslant 1$ holds for all $x\in R$, find the range of real number $k$;<br/>$(2)$ When $k=4$, let the minimum value of $f\left(x\right)$ be $m$, and let positive numbers $a$ and $b$ satisfy $a+4b=m$, find the minimum value of $\frac{1}{a}+\frac{1}{b}$.

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