
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 $f(x) = \begin{cases} \frac{1}{x} - 1, & 0 < x < 1 \\ 1 - \frac{1}{x}, & x \geq 1 \end{cases}.$  
(1) Determine the monotonicity of the function $f(x)$ on the intervals $(0, 1)$ and $[1, +\infty)$ (no proof required);  
(2) If $0 < a < b$ 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 for $x \in [a, b]$, the range of $f(x)$ is $[ma, mb]$ (where $m \neq 0$), determine the range of values of the real number $m$.

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