
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
Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels \(1, 2, \ldots, 7\), and the label on the eighth side can be changed and begins as \(1\). She rolls it several times, until each of \(1, 2, \ldots, 7\) appears at least once. After each roll, if \(k\) is the smallest positive integer that she has not rolled so far, she relabels the eighth side with \(k\). The probability that 7 is the last number she rolls is \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\).

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