
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 frog jumps from vertex to vertex in a hexagon \(A B C D E F\), moving to one of the neighboring vertices each time.

a) In how many ways can it reach vertex \(C\) from vertex \(A\) in \(n\) jumps?

b) The same question, but with the condition that it cannot jump to vertex \(D\)?

c) Suppose the frog's journey starts at vertex \(A\) and there is a mine at vertex \(D\). Each second, it makes another jump. What is the probability that it is still alive after \(n\) seconds?

d)* What is the average life expectancy of such frogs?

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