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

$(I)$ 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.

$(II)$ When $b=1$, is there a fixed point $T$ on the $x$-axis such that $\overrightarrow{TA} \cdot \overrightarrow{TB}$ is constant? Explain your reasoning.

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