
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 11 distinct positive integers \(a_{1}, a_{2}, \cdots, a_{11}\) with a total sum less than 2007. On the blackboard, the numbers 1, 2, ..., 2007 are written in sequence. A sequence of 22 consecutive operations is defined as an operation set. In the \(i\)-th operation, you can choose any number from the numbers currently on the blackboard:

When \(1 \leq i \leq 11\), add \(a_{i}\) to the chosen number.
When \(12 \leq i \leq 22\), subtract \(a_{i-11}\) from the chosen number.

If the final result is an even permutation of 1, 2, ..., 2007, the operation set is called optimal; if the result is an odd permutation, the operation set is called suboptimal.

Question: Which type of operation sets is more frequent, and by how much?

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