
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 $W: y^{2}=4x$ and the line $x=4$, which intersects the parabola at points $A$ and $B$. Let point $P(x_{0},y_{0})$ $(x_{0} < 4,y_{0}\geqslant 0)$ be a moving point on the parabola, and lines $PA$ and $PB$ intersect the $x$-axis at points $M$ and $N$, respectively.
(Ⅰ) If the area of triangle $PAB$ is 4, find the coordinates of point $P$.
(Ⅱ) When line $PA$ is perpendicular to $PB$, find the length of segment $PA$.
(Ⅲ) If the area of triangle $PMN$ is equal to the area of triangle $PAB$, find the area of triangle $PMN$.

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