
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the left and right foci as $F_{1}$ and $F_{2}$, where the coordinates of the right focus are $(1,0)$, and the eccentricity of the ellipse is $\frac{1}{2}$. <br/>$(1)$ Find the standard equation of the ellipse;<br/>$(2)$ Given that point $P(1,t)$ lies on the ellipse, a line $l$ passing through point $F_{2}$ intersects the ellipse at points $A$ and $B$ distinct from point $P$, and the area of $\triangle PAB$ is $\frac{9\sqrt{2}}{7}$, find the equation of the line $l$.

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