
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 a square $ABCD$ with side length $a$ and a segment $KL$ with length $5a$ such that $A \equiv K$ and the side $AB$ lies on the segment $KL$. The square $ABCD$ "rolls" (rotates around the bottom right corner of the square by $90^\circ$) along the segment $KL$ until the side $AB$ again aligns with part of the segment $KL(B \equiv L)$.

a) Draw the path that point $S$, which is the midpoint of side $AB$, will take.

b) Determine the length of the curve traced by point $S$ during the motion.

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