
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
Let \( n \) be a positive integer. A row of \( n+1 \) squares is written from left to right, numbered \( 0, 1, 2, \ldots, n \).

Two frogs, named Alphonse and Beryl, begin a race starting at square 0. For each second that passes, Alphonse and Beryl make a jump to the right according to the following rules: if there are at least eight squares to the right of Alphonse, then Alphonse jumps eight squares to the right. Otherwise, Alphonse jumps one square to the right. If there are at least seven squares to the right of Beryl, then Beryl jumps seven squares to the right. Otherwise, Beryl jumps one square to the right. Let \( A(n) \) and \( B(n) \) respectively denote the number of seconds for Alphonse and Beryl to reach square \( n \). For example, \( A(40) = 5 \) and \( B(40) = 10 \).

(a) Determine an integer \( n > 200 \) for which \( B(n) < A(n) \).

(b) Determine the largest integer \( n \) for which \( B(n) \leq A(n) \).

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