
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 left and right vertices $A$ and $B$, respectively, and a major axis length of $8$. Point $T$ is on the ellipse, and the product of the slopes of lines $TA$ and $TB$ is $-\frac{3}{4}$.

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

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

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