
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
Given a sequence \( 204 a_1, a_2, \cdots, a_n \) where each \( a_k \in \{0, 1\} \) for \( 1 \leqslant k \leqslant n \). For each \( k \) such that \( 1 \leqslant k \leqslant n-1 \), define a new sequence \( b_k \) such that:
- \( b_k = 0 \) if \( a_k \) and \( a_{k+1} \) are the same,
- \( b_k = 1 \) if \( a_k \) and \( a_{k+1} \) are different.

This operation generates a new sequence \( b_1, b_2, \cdots, b_{n-1} \), where each \( b_k \in \{0, 1\} \) for \( k = 1, 2, \cdots, n-1 \).

By repeating this operation, a triangle of numbers consisting of only 0s and 1s is formed, with the last row containing just a single number. Find the maximum sum of all 1s in this triangular number table.

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