
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 functions $f(x) = 1 + \frac{4}{x}$ and $g(x) = \log_{2}x$.

(1) Let the function $h(x) = g(x) - f(x)$. Find the range of $h(x)$ in the interval $[2, 4]$.

(2) Define $\min\{p, q\}$ as the smaller of $p$ and $q$. Let the function $H(x) = \min\{f(x), g(x)\}$ for $x > 0$.

   (i) Find the intervals of monotonicity and the extremum of $H(x)$.
   
   (ii) If the equation $H(x) = k$ has two distinct real roots, find the range of the real number $k$.

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