
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
In an artificial pond, there are 13 lily pads lined up from 0 to 12. A frog begins its risky journey on pad 3. When the frog is on pad \(N\), \(0 < N < 12\), the jumping probabilities are as follows: to pad \(N-1\) with probability \(\frac{(N+1)}{13}\), and to \(N+1\) with a probability \(1 - \frac{(N+1)}{13}\). If the frog lands on pad 0, a snake will eat it. If the frog reaches pad 12, it safely escapes. Find the probability that the frog escapes the snake.
- **A)** \(\frac{101}{223}\)
- **B)** \(\frac{99}{200}\)
- **C)** \(\frac{5}{11}\)
- **D)** \(\frac{120}{253}\)
- **E)** \(\frac{1}{2}\)

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