
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 parabola $C: y^2 = 4x$ with focus $F$ and directrix $l$. Circle $F$ intersects $C$ at points $A$ and $B$, and intersects the negative half of the x-axis at point $P$.
(Ⅰ) If the chord cut by $l$ on circle $F$ has a length of $2\sqrt{5}$, find $|AB|$;
(Ⅱ) Determine the number of intersection points between line $PA$ and $C$, and explain the reason.

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