
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) Convert the polar coordinates $(2,\frac{\pi }{6})$ to rectangular coordinates.

(2) Find the maximum value of the function $f(x)=\frac{x}{2}+\cos x,x\in [0,\frac{\pi }{2}]$.

(3) Find the shortest distance from the moving point $P$ on the circle $C$: $ρ=-4\sin θ$ to the line $l$: $ρ \sin (θ +\frac{\pi }{4})=\sqrt{2}$.

(4) Given that the domain of the function $f(x)$ is $(0,+∞)$, if $y=\frac{f(x)}{x}$ is an increasing function on $(0,+∞)$, then $f(x)$ is called a "first-order ratio increasing function"; if $y=\frac{f(x)}{{{x}^{2}}}$ is an increasing function on $(0,+∞)$, then $f(x)$ is called a "second-order ratio increasing function". We denote the set of all "first-order ratio increasing functions" as $Ω_{1}$, and the set of all "second-order ratio increasing functions" as $Ω_{2}$. Given the function $f(x)={{x}^{3}}-2m{{x}^{2}}-mx,$ if $f(x)∈ {{\Omega }\_{1}}$, and $f(x)∉Ω\_{2}$, the range of values for the real number $m$ is to be determined.

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