
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
Consider the function $f(x) = x^2 - mx (m > 0)$. Let its minimum value on the interval $[0,2]$ be denoted as $g(m)$.
(i) If $0 < m \leq 4$, find the explicit formula for the function $g(m)$;  
(ii) Let $h(x)$ be an even function defined on $(-\infty,0)\cup(0,+\infty)$, and when $x > 0$, $h(x) = g(x)$. If $h(t) > h(4)$, determine the range of real numbers for $t$.

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