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I have a question please can you give me a hint how to solve it. It is in thermodynamics

A vertical piston-cylinder device initially contains 0.01 m^3 of steam at 200 C. The mass of the frictionless piston is such that it maintains a constant pressure of 500 kPa inside. Now steam at 1 MPa and 350 C is allowed to enter the cylinder from a supply line until the volume inside doubles. Neglecting any heat transfer that may have taken place during the process, determine (a)the final temperature of the steam in the cylinder and (b)the amount of mass that has entered.

This is were my solution but i couldn't complete it:

State(1):500 kPa, 200 C, 0.01 m^3

State(2):500 kPa, 0.02 m^3

at the inlet: Pi=1 MPa, Ti=350 C

I've found that state 1 and the steam entered are super-heated and i got specific volume for the inlet and state 1 also the h and u from the table of super-heated water vapor

then got m1=V1/v1

m(inlet) = m2-m1

-W + (m h)inlet = m2 u2 - m1 u1....(1)

m of the inlet, m2 and u2 are unknowns

W=P(V2 - V1) = 500(0.02 - 0.01) = 5kJ

or we can write eq(1) as:

m2 h2 - m1 h1 - (m h)inlet = 0

m2, h2 and m of the inlet are unknowns

since h = Pv + u

please can you help me?