
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 \( A(n) \) denote the largest odd divisor of the number \( n \). For example, \( A(21) = 21 \), \( A(72) = 9 \), and \( A(64) = 1 \). Find the sum \( A(111) + A(112) + \ldots + A(218) + A(219) \).

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