
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 $f(x)=x+ \frac{m}{x}+2m$, where $m$ is a real constant.
(1) If the minimum distance from point $P$ on the graph of function $f(x)$ to fixed point $Q(0,2)$ is $\sqrt{2}$, find the value of $m$;
(2) If $m < 0$, find the range of $k$ such that the inequality $f(x)\leqslant kx$ has a solution when $x\in[\frac{1}{2},1]$.

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