
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 \( a_{1}, a_{2}, \ldots, a_{n} \) be numbers equal to 0 or 1. We define \( b_{1}, b_{2}, \ldots, b_{n-1} \) as follows: if \( a_{k} = a_{k+1} \), then \( b_{k} = 0 \); otherwise, \( b_{k} = 1 \). The process is repeated to fill a triangular table. What is the maximum number of 1's that this table can contain?

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