
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
(The full score of this question is 12 points)  
Given the functions $f(x) = -x^2 + 8x$, $g(x) = 6\ln x + m$.  
(1) Find the maximum value $h(t)$ of $f(x)$ in the interval $[t, t+1]$.  
(2) Does there exist a real number $m$ such that the graph of $y = f(x)$ and the graph of $y = g(x)$ have exactly three different intersection points?  
If it exists, find the range of $m$; if not, explain why.

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