
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 function $f(x) = ax^2 + \ln(x+1)$, $(a \in \mathbb{R})$.
(I) Suppose the function $Y = F(X-1)$ has domain $D$
    - Find the domain $D$.
    - If the function $h(x) = x^4 + [f(x) - \ln(x+1)]\left(x+\frac{1}{x}\right) + cx^2 + f'(0)$ has a zero in $D$, find the minimum value of $a^2 + c^2$.
(II) When $a = \frac{1}{2}$, $g(x) = f'(x-1) + bf(x-1) - ab(x-1)^2 + 2a$, if for any $x \in [1, e]$, $\frac{2}{e} \leq g(x) \leq 2e$ holds true, find the range of real numbers $b$. (Note: $e$ is the base of the natural logarithm.)
(III) When $x \in [0, +\infty)$, the points on the graph of the function $y = f(x)$ are all within the planar region represented by $\begin{cases} x \geq 0 \\ y - x \leq 0 \end{cases}$, find the range of the real number $a$.

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