
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 the line $y=\frac{1}{2}x$ intersects the hyperbola $y=\frac{k}{x}$ ($k>0$) at points $A$ and $B$, and the abscissa of point $A$ is $4$. Another line $l$ passing through the origin $O$ intersects the hyperbola $y=kx$ ($k>0$) at points $P$ and $Q$ ($P$ is in the first quadrant). If the quadrilateral formed by points $A$, $B$, $P$, and $Q$ is a rectangle, then the coordinates of point $P$ are ______.

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