
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
Define a sequence \( a_{i, j} \) of integers such that \( a_{1, n} = n^n \) for \( n \geq 1 \) and \( a_{i, j} = a_{i-1, j} + a_{i-1, j+1} \) for all \( i, j \geq 1 \). Find the last (decimal) digit of \( a_{128, 1} \).

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