
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), whose left and right vertices are $A$ and $B$ respectively, and the length of the major axis is $8$. Let $T$ be any point on the ellipse, and the product of the slopes of lines $TA$ and $TB$ is $-\frac{3}{4}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Let $O$ be the origin, and a moving line passing through point $M(0,2)$ intersects the ellipse $C$ at points $P$ and $Q$. Find the range of $\overrightarrow{OP} \cdot \overrightarrow{OQ} + \overrightarrow{MP} \cdot \overrightarrow{MQ}$.

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