
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 a function $f(x) = kx$ ($k \neq 0$), and it satisfies $f(x + 1) \cdot f(x) = x^2 + x$,  
(I) Find the explicit formula for the function $f(x)$;  
(II) Suppose $f(x)$ is an increasing function on $\mathbb{R}$ and let $h(x) = \frac{f(x) + 1}{f(x) - 1}$ (where $f(x) \neq 1$), is there a real number $m$ such that both the domain and range of $h(x)$ are $[m, m+1]$? If such $m$ exists, find its value; if not, please explain why.

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