
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 $f(x)=|2x|+|2x-1|$.
$(1)$ Find the solution set of $f(x)\geqslant 5$;
$(2)$ Let $M$ be the minimum value of the function $f(x)$. If $a$, $b\in \mathbb{R}^{+}$ and $2a+b=M$, find the minimum value of $\frac{1}{a}+\frac{2}{b}$.

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