
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 that the major axis of an ellipse is $2a$, the foci are $F_{1}(-\sqrt{3}$,$0)$ and $F_{2}(\sqrt{3}$,$0)$, the distance from point $F_{1}$ to the line $x=-\frac{a^{2}}{\sqrt{3}}$ is $\frac{\sqrt{3}}{3}$, and a line $l$ passing through point $F_{2}$ at an angle of $45^{\circ}$ intersects the ellipse at points $A$ and $B$. <br/>$(1)$ Find the equation of the ellipse;<br/>$(2)$ Find the length of segment $AB$.

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