
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 $f(x) = \ln{x}$ and $g(x) = \frac{2a}{3}x^3 + 2(1-a)x^2 - 8x + 8a + 7$.
(1) Find the range of function $g(x)$ on the interval $[0, 3]$ if the equation of the tangent line to the curve $y = g(x)$ at the point $(2, g(2))$ is $y = ax - l$.
(2) Define function $h(x)$ as follows when $x > 0$:

$$
h(x) =
\begin{cases}
f(x), & \text{if } f(x) < g(x) \\
g(x), & \text{if } f(x) \geq g(x)
\end{cases}
$$

Find the range of real number values for $a$ if the function $y = h(x)$ has three zero points.

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