Careless Astronauts

https://isaacphysics.org/questions/rev_astronaut?board=a7dbbcb5-4786-4150-944b-febdcc30d92c&stage=further_a
For this question I dont really know how youre supposed to come up with the conclusion that the answer is 0.
I initially tried using angular momentum but I didnt get anywhere and dont really think its relevant (atleast according to the video), and using the video hint got:
m*OMEGA^2*L*sin(theta) = 2Tsin(theta)
2m*omega^2*L*cos(theta)= 2Tcos(theta)
but I dont really know what Im supposed to from here.
Also could someone explain why OMEGA > omega - I dont really understand that part?

So after a while of spinning both astronauts have the same w? Instead of the heavier one having w and the lighter one having
Ω?
I get why the lighter astronaut gets pulled to the center (I think)
initially their centripetal force is greater than the amount required to keep them rotating around the com (as shown in the video hint), so they go towards the com.
This means that the heavier astronauts get pushed away from the centre (until theta=45 degrees, from which they go towards the centre again)
So because you get a nonsensical result the only explanation is sin(theta) = 0 so theyve reached the centre?

Sorry! I was visualising it wrong - theres nothing special about theta being 45 degrees (other than it being the starting angle).
I originally thought that what they meant by the lighter astronaut having W was that since its distance to the centre decreased its moment of inertia I=mr^2 should decrease which means its "easier" to rotate and spins faster (to keep angular momentum L=Iw constant) and vice versa for the heavier astronaut.
But I dont think that really makes sense since the whole system rotates with a constant w (atleast after the rotation becomes stable) so how can different parts of the system have different angular speeds?

I got a reply from them if youd like to see
I basically asked why the video uses omega and OMEGA and what purpose does conserving angular momentum have in this case (the equation is given in hint 4)?
"I think hat the two omegas are there since the system is not in equilibrium at that stage. I think your argument for theta looks reasonable. Yes, it is a more quantitative way of seeing what changes must occur.
You can't do this from an angular momentum point of view. What you can do is say that the angular momentum from the moment the tools have been released will be conserved. Then you can connect the final rate of rotation of the two astronauts at distance L from the centre and the angular velocity of the four astronauts when the tools have been lost. A factor of three I think. But you can't say how the configuration of the system will change by using angular momentum. That requires applying forces whereas the angular momentum responds to torques.
You have made some good points in this question."