
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 these rules:
- if there are at least eight squares to the right of Alphonse, he jumps eight squares to the right; otherwise, he jumps one square to the right.
- if there are at least seven squares to the right of Beryl, she jumps seven squares to the right; otherwise, she 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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