
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
A circle with radius $R$ has a tangent at point $A$ and a chord $MN$ parallel to the tangent, at a distance $x$ from the tangent. Let $P$ be the projection of point $M$ on the tangent, and $Q$ be the projection of point $N$ on the tangent. Determine the value of $x$ such that the perimeter of the rectangle $PMNQ$ is $2k$, and establish the condition for the possibility of a solution as well as the number of solutions.

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