OK ... I'm stuck ... again
I assume that when you say to write down a conversion reaction with stoichiometric amount of water assuming a complete conversion, you mean the following:
CH4 + 2O2 -> C02 + 2H2O
But I'm not sure what you mean by taking into account the efficiency.
In the next step you talked about adding water to both sides of the reaction; since the problem states that the initial CH4:H2O ratio was 1:3, I assume that the formula becomes:
CH4 + 2O2 + 3H2O -> CO2 + 5H2O
And then I also didn't understand the last step ... sorry