
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 tourist gets lost in a forest; he knows the shape of the forest but does not know his current location.

a) If the forest is a circle with diameter \( d \), moving in any fixed direction, the tourist will inevitably exit the forest by traveling a distance \( d \) because any segment of length \( d \) starting inside the forest will intersect its boundary.

b) If the forest is a half-plane and the tourist knows he is at a distance \( d \) from the boundary (but not in which direction the boundary lies), he can exit the forest by first moving a segment \( AB \) of length \( d \) in any direction, and then circumnavigating a circle \( S \) with center at the starting point \( A \) and radius \( d \). This path of length \( d + 2\pi d \) will likely lead him to the forest boundary located somewhere \( d \) away from point \( A \).

Can the tourist choose another path that will certainly lead him out of the forest in a distance shorter than \( (1+2\pi) d \)?

It is relatively easy to find the optimal path for exiting the forest in the shortest time in the case of a circular forest, but it is not always so simple for other shapes of forests. Mathematical studies have shown the most advantageous path for a sufficiently elongated rectangle with an aspect ratio greater than 2.278, but it is not yet known what the optimal path would be for other shapes like squares or equilateral triangles.

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