
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
In the Cartesian coordinate system, let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C$: $\frac{x^2}{4}+y^2=1$. <br/>$(1)$ If $T$ is any point on the ellipse $C$, find the range of $\overrightarrow{TF_{1}} \cdot \overrightarrow{TF_{2}}$; <br/>$(2)$ Let $A(0,-1)$, the line $l$ intersects the ellipse $C$ at points $B$ and $D$. If $\triangle ABD$ is an isosceles right triangle with $A$ as the right angle vertex, find the equation of the line $l$.

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