
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
At the start of this problem, six frogs are sitting with one at each of the six vertices of a regular hexagon. Every minute, we choose a frog to jump over another frog using one of the two rules illustrated below. If a frog at point \( F \) jumps over a frog at point \( P \), the frog will land at point \( F' \) such that \( F, P \), and \( F' \) are collinear and:
- using Rule 1, \( F' P = 2FP \).
- using Rule 2, \( F' P = \frac{FP}{2} \).

It is up to us to choose which frog to take the leap and which frog to jump over.
(a) If we only use Rule 1, is it possible for some frog to land at the center of the original hexagon after a finite amount of time?
(b) If both Rule 1 and Rule 2 are allowed (freely choosing which rule to use, which frog to jump, and which frog it jumps over), is it possible for some frog to land at the center of the original hexagon after a finite amount of time?

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