
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
If the coordinates of the two foci of ellipse $C$ are $F_{1}(-1,0)$ and $F_{2}(1,0)$, and it passes through the point $P(1,\frac{3}{2})$, a perpendicular line to the $x$-axis is drawn from $F_{2}$ intersecting the ellipse at points $A$ and $B$. <br/>$(1)$ Find the standard equation of the ellipse $C$; <br/>$(2)$ Find the area of triangle $ABF_{1}$.

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