Tough Problem

There are 422 possible values of k.

Any help?

The question is ambiguous.  Does it mean

1.Look for a value of k only using (2a-b,a); then look for a separate value of k using only (2a_b+2,a); then look for a separate value of k using (a+2b+2,b)

or

2. Look for values of k using one of the pairs selected at random at each stage

or

3. Something else?

If 1, then there is only one value of k, namely 2013. (the patterns become clear if you just work out a few of the initial stages for each expression).