
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 $f(x)=\begin{cases} \frac{1}{x}-1, & 0 < x < 1 \\ 1-\frac{1}{x}, & x\geqslant 1 \end{cases}$.

(1) Determine the monotonicity of the function $f(x)$ in the intervals $(0,1)$ and $[1,+\infty)$ (no proof required);

(2) When $0 < a < b$, and $f(a)=f(b)$, find the value of $\frac{1}{a}+\frac{1}{b}$;

(3) If there exist real numbers $a,b$ ($1 < a < b$) such that when $x\in [a,b]$, the range of $f(x)$ is $[ma,mb]$ ($m\ne 0$), find the range of the real number $m$.

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