
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 $f(x) = -x^3 + ax^2 + 1$ where $a \in \mathbb{R}$,
1. If there is a tangent line at the point on the graph of $f(x)$ with x-coordinate $\frac{2}{3}$ that is vertical to the y-axis, find the value of $a$.
2. If $f(x)$ has two distinct extreme points in the interval $(-2, 3)$, find the range of values for $a$.
3. Under the condition in (1), determine if there exists a real number $m$ such that the function $g(x) = x^4 - 5x^3 + (2-m)x^2 + 1$ has exactly three points of intersection with the graph of $f(x)$. If there is such an $m$, find its value; if not, explain why.

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