For non-negative integers \(a \) and \(b\), Let \(T_i \) denote the \(i^{\text{th}} \) largest positive integer such that there's no solution to \( 11a + 12b = T_i \).

What is the value of \(T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7 + T_8 + T_9 + T_{10} \)?

**Details and assumptions:**

- This is a Computer Science problem, but you're welcome to try to solve it with mathematics alone.

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