
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]


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After studying, it was found that the graph of all cubic functions of the form $f(x) = ax^3 + bx^2 + cx + d (a \neq 0)$ has a center of symmetry. Let $f'(x)$ be the derivative of the cubic function $y = f(x)$, and $f''(x)$ be the derivative of the function $f'(x)$. If the equation $f''(x) = 0$ has a real root $x_0$, then $(x_0, f(x_0))$ is called the center of symmetry of the graph of the cubic function $y = f(x)$. Based on the above information and related knowledge, answer the following questions: Given the function $f(x) = 2x^3 - 3x^2 + 3x - 2$.

$(1)$ Find the center of symmetry of the graph of the function $f(x)$ and the value of $f(\frac{1}{2024}) + f(\frac{2}{2024}) + \cdots + f(\frac{2023}{2024})$.

$(2)$ If $a > 0$, solve the inequality in terms of $x$: $f'(x) > (3a + 6)x^2 - (a + 9)x + 4$.

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