
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 a focal length of $4$, and the ratio of its major axis to its minor axis is $\sqrt{3} : 1$.

$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Let $F$ be the right focus of the ellipse $C$, and $T$ be any point with a non-zero ordinate on the line $x=t (t \in \mathbb{R}, t \neq 2)$. A perpendicular line from $F$ to $TF$ intersects the ellipse $C$ at points $P$ and $Q$. If the line $OT$ (where $O$ is the origin) bisects the line segment $PQ$, find the value of $t$.

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