
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, and its left and right foci are $F_1$ and $F_2$, respectively. A line passing through $F_1$ intersects the ellipse $C$ at points $A$ and $B$.

$(1)$ If the circle with diameter $AF_1$ is tangent to the circle $x^2 + y^2 = 9$, find the length of the major axis of the ellipse;

$(2)$ When $b = 1$, is there a fixed point $T$ on the $x$-axis such that $\overrightarrow{TA} \cdot \overrightarrow{TB}$ is a constant value? If it exists, find the constant value; if not, explain why.

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