
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
A decimal number in the form of \(0. a_{1} a_{2}^{(k)} \cdots a_{n}^{(k)} \cdots \) (denoted as \(A(a_{1}, k)\)), where the digit \(a_{1}\) is any natural number from 1 to 9, has its subsequent digits determined as follows: \(a_{2}^{(k)}\) is the unit digit of the product \(k a_{1}\), and \(a_{n}^{(k)}\) is the unit digit of the product \(k a_{n-1}^{(k)}\) for \(n = 3, 4, \cdots, k\). Find the sum \(\sum_{k=1}^{9} \sum_{a_{1}=1}^{9} A(a_{1}, k)\).

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