
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 $f(x) = \frac{3}{8}x^2 + \ln x + 2$ and $g(x) = x$.
(I) Find the extreme points of the function $F(x) = f(x) - 2g(x)$.
(II) If the function $F(x) = f(x) - 2g(x)$ has a zero point on the interval $(e^t, +\infty)$ where $t \in \mathbb{Z}$, find the maximum value of $t$.
(III) If $b_n = g(n)^{\frac{1}{g(n+1)}}$ where $n \in \mathbb{N}^*$, does there exist $b_n = b_m$ (with $m \neq n$) in the sequence $\{b_n\}$? If such terms exist, find all pairs of equal terms; if not, explain why. (e is the base of the natural logarithm, approximately equal to 2.718).

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