This analysis is based on a false premise. Regardless of the efficiency of a steam boiler, and regardless of how much energy it takes to run an engine on steam, the heat is already being made in the internal combustion engine and being wasted out the exhaust pipe. Put one drop of water in the cylinder and it will add kinetic energy to the piston.
it's easy to say "based on false premise"
I haven't seen proof yet

Frankly I don't know.
I cannot say a six stroke is better or worse than a four stroke;
I simply can't.
Think about it this way:
if you add two strokes as intended, the fourth would be removing the pressure from the cylinder via your exhaust pipes.
The hottest part is the burned fuel, the hot gases are exhaled into the exhaust and muffler (both being heated no matter what follows next in the cylinder)
So, the heat peaked already, the cylinder is starting to cool down.
You add a water cycle and inject water into the cylinder...
Great...
Fifths stroke questions...
Now how much heat is left, how many joules of thermal energy are you able to transform into kinetic energy (turning water into steam) within this one stroke?
(within the ~0.017 seconds you have /based on 3600 rpm)
And, what is the exact amount of water that is transformed for that reason.
How much pressure can that much water produce without collapsing into water again, is that enough to actually move the piston down to BDP or close?
Now out with the steam.
Okay passive stroke, no big deal
.... you think!
You cooled the cylinder by a certain amount, what amount?
Will the steam condense at any one point in the cylinder (maybe the spark plug gap?)
Are any sensitive materials nearby, will the temperature still be high enough to get the next combustion as powerfull as the previous (and yes.. that IS important.. really important!)
Just think you shortened out the spark plug.. no spark no running engine;
or you loose combustion energy with every single stroke..
no matter how high the energy was with your first combustion
At one time there will be no more energy left and all you do is fill your cylinder with water
To raise the initial energy level you will need to add fuel and air, quite the opposite of what your goal was.
(basically what you do when starting an engine cold in the winter.. much more fuel is needed than a engine that ran for 45min in the summer

)
And what I think has been forgotten already..
the main heat source (the combusted gas) is already gone, it heated the muffler, that energy is lost if you tap into the cylinder.
Well, you can run a steam pipe through the exhaust pipes and muffler and run a generator from that, but that's not a six stroke is it?
So...
if it'd be that easy to convert, then I do not understand why the big companies (not even the ones that are proud of energy saving engines) came up with an engine like that.
*shrugs*
I've seen many different engine layouts five stroke even
and all claim to be more eco friendly and fuel saving and whatnot.
Yet I don't know any car with one of those...
I really like to see someone trying, I really do,
I wish you (whoever wants to try) the best of luck, and the biggest possible success; honestly.
But I doubt there will be an effect that's even noteworthy
'sid