
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
$(1)$ Given $x$ and $y$ satisfy $x \gt 0$, $y \gt 0$, and $\frac{2}{x}+\frac{1}{y}=1$, find the minimum value of $x+2y$;<br/>$(2)$ Given $x \lt 3$, find the maximum value of $f(x)=\frac{4}{{x-3}}+x$;<br/>$(3)$ Let $m$ and $n$ be positive real numbers such that $m+n=2$, find the minimum value of $\frac{n}{m}+\frac{1}{{2n}}$.

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