Equilbrium temp of ice and steam mixture

1. The problem statement, all variables and given/known data
You are asked to mix equal masses of of ice and steam at 0.00°C and 100.00°C respectively, What will the final temperature of the mixture be? The mixture is in a perfect insulator.

ms=mass of steam
mi=mass of ice
LHV,s=Heat of vaporization of steam=2257J/g
LHF,i=Heat of fusion of ice=334J/g
Ti,s=100°C=373K
Ti,i=0.00°C=273K
cw=4.19J/(g*K)

2. Relevant equations
Q=mL
Q=mcΔT

3. The attempt at a solution

I am fairly confident that my thought process is correct. I just keep making algebra mistakes(I assume) but I can’t find what my error is. Please let me know where my process breaks down please.

I did this in 3 parts

Part 1)
So first I consider the situation.
Since the mixture is in a perfect insulator, that means no heat is lost to the surroundings.

Q=0

That means that the heat lost by the steam is gained by the ice. Q for the steam will be negative since it is losing heat.

Q=0=Qgained+Qlost=Qgained+(-Qlost)

Qlost=Qgained

From here, the Qlost is given by the sum of energy used to turn steam into water, then to bring the steam-turned-water to thermal equilibrium

Qlost=mscw(Ti,s-Teq)+msLHV

and Qgained is given by the sum of energy used to melt the ice, then to bring the ice-turned-water to thermal equilibrium.

Qgained=micw(Teq-Ti,i)+miLHF

Again, since there is no heat lost to the surroundings, and the mixture is allowed to reach thermal equilibrium then…

Qlost=Qgained

Part 2)
Now we plug in to be able to solve for Teq

mscw(Ti,s-Teq)+msLHV=micw(Teq-Ti,i)+miLHF

mscwTi,s-mscwTeq+msLHV=micwTeq-micwTi,i+miLHF

Moving all terms containing Teq to one side

mscwTi,s+micwTi,i+msLHV-miLHF=micwTeq+mscwTeq

Factor out Teq

mscwTi,s+micwTi,i+msLHV-miLHF=(micw+mscw)Teq

and finally solving for Teq results in

Teq=mscwTi,s+micwTi,i+msLHV-miLHF/(micw+mscw)

We can do one more thing, since the masses are equal, we can factor out the mass.

Teq=mcwTi,s+mcwTi,i+mLHVmLHF/(mcw+mcw)

Teq=m(cwTi,s+cwTi,i+LHV-LHF)/m(cw+cw)

Teq=(cwTi,s+cwTi,i+LHV-LHF)/2cw

Part 3)
Plug in known values

Teq=[(4.19J/(g*K)(373K)+(4.19J/(g*K))(273K)+(2257J/g)-(334J/g)]/[(2*4.19J/(g*K))]

Teq=552K Obviously incorrect

Please let me know where I am making my mistake. I seperated my work into 3 parts to make it easier to point out where i made an error. Thank you I have a test on Friday I NEEEEEED to understand how to do this properly. Thank you

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