
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 $$f(x)=|2- \frac {1}{x}|(x＞0)$$.
(1) When $0＜a＜b$ and $f(a)=f(b)$, ① find the value of $$\frac {1}{a}+ \frac {1}{b}$$; ② find the range of $$\frac {1}{a^{2}}+ \frac {2}{b^{2}}$$;
(2) Suppose the domain of function $g(x)$ is $D$, if there exists an interval \[m，n\]⊆D, when $x∈\[m，n\]$, the range of $g(x)$ is \[m，n\], then the function $g(x)$ is called a "range-preserving function" on $D$, and the interval \[m，n\] is called an "equal-range interval". Determine whether the function $f(x)$ is a "range-preserving function" on $(0，+∞)$? If so, find its "equal-range interval"; if not, please explain why.

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