
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
We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ squares. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, let $N(k)$ denote the number of ways to cut the board such that a cut goes along the edge $k$. Calculate the last digit of the sum obtained when all $N(k)$ are added, where $k$ is an inner edge.

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