
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
Using the digits \(1, 2, 3, 4, 5, 6, 7, 8\), we can form \(8!\) \((= 40320)\) 8-digit numbers in which the eight digits are all distinct. For \(1 \leq k \leq 40320\), let \(a_k\) denote the \(k\)-th number if these numbers are arranged in increasing order:
\[ 12345678, 12345687, 12345768, \ldots, 87654321 \]
that is, \(a_1 = 12345678\), \(a_2 = 12345687\), \ldots, \(a_{40320} = 87654321\). Find \(a_{2099} - a_{2001}\).

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