
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
Given \( a_1, a_2, \cdots, a_{11} \) as 11 distinct positive integers such that their sum is less than 2007. On the board are the numbers \( 1, 2, \cdots, 2007 \). A sequence of 22 operations is defined as follows: in the \( i \)-th operation, you can choose any number on the board. For \( 1 \leqslant i \leqslant 11 \), add \( a_i \). For \( 12 \leqslant i \leqslant 22 \), subtract \( a_{i-11} \). If the final result is an even permutation of \( 1, 2, \cdots, 2007 \), this sequence of operations is called "excellent." If the final result is an odd permutation, it is called "sub-optimal." Which type of sequences—excellent or sub-optimal—are more numerous, and by how much?

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