
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 functions $f(x)=x+ \frac{a}{x},a\in R,g(x)=x^{2}-2mx+2,m\in R$
(1) Determine the monotonicity of $f(x)$ on $(0,+\infty)$ when $a < 0$;
(2) When $a=-4$, for any real number $x_{1}$, $x_{2}\in[1,2]$, there is always $f(x_{1})\leqslant g(x_{2})$, find the range of values of the real number $m$;
(3) When $m= \frac{3}{2}$, $F(x)= \begin{cases} \overset{f(x),x < \frac{1}{2}{且}x\neq 0}{g(x),x\geqslant \frac{1}{2}}\end{cases}$, $y=|F(x)|$ is monotonically decreasing on $(0,1)$, find the range of values of $a$.

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