
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 point $F$ is the focus of the parabola $C:y^{2}=4x$, and point $P(-1,1)$, the line $PF$ intersects the parabola $C$ at points $A$ and $B$, and point $D$ is the intersection of the directrix $l$ and the $x$-axis.

$(1)$ Find the equation of line $PF$;

$(2)$ Find the area $S$ of $\triangle DAB$

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