
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 $F$ is the focus of the parabola $C$: $y^{2}=2x$, point $E$ is on the ray $l$: $x=-\frac{1}{2}(y\geqslant 0)$, the perpendicular bisector of segment $EF$ intersects with $l$ at point $Q(-\frac{1}{2}, \frac{3}{4})$, and with parabola $C$ at point $P$, then the area of $\triangle PEQ$ is _______.

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