# How can I get a single solution to a set of inequality constraints?

I have a set of constraints to which I require integer solutions. However, instead of needing all possible integer solutions (as obtained by using Solve[constraints,Variables,Integers]), I need any one solution only.

Here is a specific example:

variables = {
Subscript[x, 11], Subscript[x, 12], Subscript[x, 13],
Subscript[x, 14], Subscript[x, 15], Subscript[x, 16]
};

constraints =
273 + 4 Subscript[x, 11] + Subscript[x, 12] >= 0 &&
-693 - 8 Subscript[x, 11] - 3 Subscript[x, 12] + Subscript[x, 13] >= 0 &&
483 + 4 Subscript[x, 11] + 3 Subscript[x, 12] - 3 Subscript[x, 13] +
Subscript[x, 14] >= 0 &&
-Subscript[x, 12] + 3 Subscript[x, 13] - 3 Subscript[x, 14] + Subscript[x, 15] >= 0 &&
-Subscript[x, 13] + 3 Subscript[x, 14] - 3 Subscript[x, 15] + 24 Subscript[x, 16] >= 0 &&
-Subscript[x, 14] + 3 Subscript[x, 15] - 32 Subscript[x, 16] >= 0 &&
-Subscript[x, 15] + 8 Subscript[x, 16] >= 0 &&
Subscript[x, 16] >= 0 &&
Subscript[x, 11] >= 0 &&
462 + 11 Subscript[x, 11] + Subscript[x, 12] + Subscript[x, 13] + Subscript[x, 14] +
Subscript[x, 15] - 15 Subscript[x, 16] >= 0 &&
2709 + 42 Subscript[x, 11] + 8 Subscript[x, 12] + 4 Subscript[x, 13] -
4 Subscript[x, 15] + 40 Subscript[x, 16] >= 0 &&
4536 + 54 Subscript[x, 11] + 18 Subscript[x, 12] - 2 Subscript[x, 13] -
6 Subscript[x, 14] + 6 Subscript[x, 15] - 45 Subscript[x, 16] >= 0 &&
-1134 - 27 Subscript[x, 11] - 12 Subscript[x, 13] + 8 Subscript[x, 14] -
4 Subscript[x, 15] + 24 Subscript[x, 16] >= 0 &&
-6318 - 81 Subscript[x, 11] - 27 Subscript[x, 12] + 9 Subscript[x, 13] -
3 Subscript[x, 14] + Subscript[x, 15] - 5 Subscript[x, 16] >= 0 &&
273 + 4 Subscript[x, 11] + Subscript[x, 12] == Subscript[x, 16] &&
-693 - 8 Subscript[x, 11] - 3 Subscript[x, 12] + Subscript[x, 13] == Subscript[x, 11];

solutions = Solve[constraints, variables, Integers]

I get an output, but it is very long as there are many solutions. Consequently, the runtime is significant. Hence, I want to modify Solve[] as to stop evaluating after one solution has been found. I believe that would not run for as long as the general case.