Just for a comment;
A very interesting question. It appears to be paradoxical.
Buy, in your problem, conditions are given for friction and the mass of the piston, but there are no constraints for heat transfer in and out or temperature change. So it is an incomplete problem; thanks to this imperfection, it seems to me that there can be a reasonably possible escape route. Conveniently, there is no mention of the pressure inside and outside the syringe being "different" even after the pin has been removed.
If we now assume that the gas inside the syringe is an ideal gas, then the course of $P_{gas}$ as a function of $V$ satisfies $P_{gas}(V)V=nRT$, so,
$$P_{gas}(V) = nRT/V$$
, but there is no assumption at all that the temperature does not change.
So what would $P_{gas}(V)$ be if we assume a limit where the cylinder gains no energy at all, i.e. so slow that it gains no kinetic energy? In order for this to be the case, it must always be the case that $P_{gas} dV = P_{ext} dV$, so
$$P_{gas}=P_{ext} $$
shall be always satisfied. The only way to force it to do this is to change the temperature conveniently; if we carry even a small amount of energy into the kinetic energy of the syringe because of the following "withdrawn statement".
That is, the temperature of the gas in the syringe can and must be applied as a function of $V$ in the following form;
$$P_{gas}(V)=P_{ext} = nRT_{gas}(V)/V$$
So,
$$T_{gas}(V)=P_{ext} V/nR$$
That means, if the temperature in the syringe can be varied in this way, a physically possible situation can be artificially created.
◆The following statement is withdrawn on 28 April 2021:
Let S be the cross-sectional area of the piston.
When the gas inside the syringe expands (i.e. $P_{gas}>P_{ext}$), the piston receives a force of $F_1=P_{gas}S$ from the gas inside the syringe in the direction of the gas expansion.
On the other hand, in this case, the sringe receives a force of $F_2=-P_{ext}S$ from the gas outside the syringe in the direction of gas expansion.
Therefore, the combined force received by the syringe is
$F_{sir} = F_1 + F_2 = (P_{gas} - P_{ext})S$ in the direction of the gas expansion.
Therefore, when moving a small distance $dl$, the syringe obtains the following energy;
$$F_{sir}\ dl = (P_{gas} - P_{ext})S\ dl = (P_{gas} - P_{ext})\ dV$$
So, during your process, the energy the syringe obtains from the gasses will be;
$${W}_{sir}=\int_{V_1}^{V_2} (P_{gas} - P_{ext}) dV$$
If there is no friction or force braking the piston, then the difference between the $W_{ext}$ and the ${W}_{piston}$ (that is ${W}_{sir}$)seems to have nowhere to go but the kinetic energy of the cylinder.
In such a situation. If the mass of the cylinder is infinitely zero, wouldn't the velocity of the cylinder be infinitely large? It seems to me that this would be a physically impossible situation.