
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 along the vertices of a hexagon $A B C D E F$, each time moving to one of the neighboring vertices.

a) In how many ways can it get from $A$ to $C$ in $n$ jumps?

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

## Frog-Minesweeper.

c) Let the frog's path start at vertex $A$, and there is a mine at vertex $D$. Every second, the frog makes a jump. What is the probability that it will still be alive after $n$ seconds?

d)* What is the average lifespan of such frogs?

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