
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
The integers \(a_{1}, a_{2}, \ldots, a_{10}\) are all greater than 1 and their sum is 2006. What is the smallest possible value of the sum

\[
\binom{a_{1}}{2} + \ldots + \binom{a_{10}}{2}?
\]

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