
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 $f(x)=\log _{\frac{1}{2}}x$, when point $M(x,y)$ moves on the graph of $y=f(x)$, point $N(x-2,ny)$ moves on the graph of the function $y=g_{n}(x)$ $(n∈N^{∗})$.
(1) Find the expression of $y=g_{n}(x)$;
(2) If the equation $g_{1}(x)=g_{2}(x-2+a)$ has real roots, find the range of values of the real number $a$;
(3) Given $H_{n}(x)=2^{g_{n}(x)}$, the range of the function $F(x)=H_{1}(x)+g_{1}(x)(0 < a\leqslant x\leqslant b)$ is $[\log _{2} \frac{ \sqrt[5]{2}}{b+2}, \log _{2} \frac{ \sqrt[4]{2}}{a+2}]$, find the values of the real numbers $a$ and $b$.

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