
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 the line $l:\begin{cases} x=1+t\cos \alpha \\ y=t\sin \alpha \end{cases}$ (where $t$ is a parameter, and $\alpha$ is the inclination angle), establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the curve $C$ is $\rho = \frac{\cos \theta }{\sin^{2} \theta }$.

(1) Find the rectangular equation of $C$ and the rectangular coordinates of its focus $F$;

(2) Given point $P(1,0)$, if line $l$ intersects $C$ at points $A$ and $B$, and $\frac{1}{|PA|}+\frac{1}{|PB|}=2$, find the area of $\triangle FAB$.

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