
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
Consider parabola $C: y^{2}=2px (p > 0)$ with focus $F$ and directrix $l.$ Suppose there is a circle centered at $F$ with radius $4$, and it intersects the directrix at points $A$ and $B$. Let $E$ be one of the intersection points between the circle and the parabola $C$ such that $\angle EAB = 90^{\circ}$.

$(1)$ Find the value of $p$.

$(2)$ Suppose a point $P$ on the parabola $C$ has an ordinate of $-1$. Points $Q$ and $R$ are two distinct points on $C$, different from $P$, such that the sum of the slopes of lines $PQ$ and $PR$ is $-1$. Determine whether the line $QR$ passes through a fixed point, and if so, find the coordinates of the fixed point. If not, explain why.

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