
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
There are 8 lily pads in a pond numbered \(1, 2, \ldots, 8\). A frog starts on lily pad 1. During the \(i\)-th second, the frog jumps from lily pad \(i\) to \(i+1\), falling into the water with probability \(\frac{1}{i+1}\). The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n) = 1\). Find \(100m + n\).

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