
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
Arc $DE$ is a quarter-circle with center $F$. The shaded region $DEF$ is "rolled" along a straight board $RS$ until it reaches its original orientation for the first time with point $F$ landing at point $F^{\prime}$. If $EF = \frac{1}{\pi}$ cm, what is the length of the path that point $F$ travels? Express your answer in simplest form.

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