
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 $f(x)=x+||x-a||$, which satisfies both $(f-2)^4$ and $f\leqslant 4$. Find the minimum value of $f(x)$. If $\frac{1}{m}+\frac{2}{n}=M (m,n\in\mathbb{Z}^{+})$, find the minimum value of $m+2n$.

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