Adiabatic expansion, irreversible, in a container?

I’m reading D. F. Lawden, Principles of Thermodynamics and Statistical Mechanics, and am trying to understand the first worked problem they have. Below is the problem and start of their solution.

My problem in understanding is this: In the problem below, I am taking a "container" to be a fixed-volume container, and I don’t know what it means if it isn’t. I am understanding the container being a fixed-volume entity, and that the gas escapes into a variable-volume chamber with a piston. But that doesn’t make sense with the explanation given below. Can someone explain a bit more what it means to have a container change volume while gas escapes into the piston chamber? Is the container variable-volume, but always at equilibrium with pressure in the surrounding environment (that is, the environment pressure changes to match the container pressure as the container pressure changes, and so the container can then be a variable volume entity? There’s no picture with the problem, and I may fundamentally misunderstand what a "container" is here and that it is a fixed-volume object (obviously not from the problem, but I want to make sure I have the right picture of what that means).

D. F. Lawden, Principles of Thermodynamics and Statistical Mechanics,
pg. 17, Problem 1:

$V_0$ moles of an ideal gas are contained in an insulated chamber at
pressure $p_0$ and temperature $T_0$. The gas slowly escapes through a
valve into an insulated cylinder provided with a frictionless piston
to which an external pressure $p_1$ ($< p_0$) is applied. Initially the
volume enclosed iby the piston is zero. When the piston comes to
rest, calculate the number of moles of gas left in the chamber and its
temperature. Find, also, the temperature of the gas in the cylinder.

Solution

The expansion of the gas is adiabatic, but not quasi-static, since
thee will be a pressure gradient across the valve until the chamber
has been reduced to $p_1 $at the end of the process. However, the
expansion of that portion of the gas which never leaves the chamber
will be quasi-static, and must be governed by [$pV^gamma$ = constant,
governing adiabatic expansion of an ideal gas by a quasi-static
process]. Thus, if $V_0, V_1$ are its volumes at the beginning and end
of the process

$p_0 V_0^gamma = p_1 V_1^gamma$

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