
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
For each positive integer \( k \) with \( n^2 \) decimal digits (and leading digit non-zero), let \( d(k) \) be the determinant of the matrix formed by writing the digits in order across the rows (so if \( k \) has decimal form \( a_1a_2 \ldots a_{n^2} \), then the matrix has elements \( b_{ij} = a_{n(i-1)+j} \)). Find \( f(n) = \sum d(k) \), where the sum is taken over all \( 9 \cdot 10^{n^2-1} \) such integers.

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