Imagine there is a painter, weighing $180$ $lbs$, that is working from a bosun's chair hung down the side of a tall building.

Suppose that he pulls down on a fall rope with such a force that he presses against the chair with a force of $100$ $lbs$. You can assume that the chair's weight is $30$ $lbs$.

For finding the acceleration of the painter and the chair, I took into account that the weights of the painter and the chair are 180 lbs and 30 lbs respectively. I used this idea to perform the following step:

$$\text {Total mass of the painter and the chair} = \left(\frac{(180 + 30) \rm lbs}{g}\right) $$

He exerts a downward force of 100 lb on the chair. His net motion will be upwards.

I think the 100 lbs force the person exerts on the chair is transferred to the rope he is pulling on.

But that is just the string he is pulling on. The diagram shows that only one end of the rope is attached to the bosun chair. That end will have an upwards force of (100 + 180 + 30) lbs (as shown in the picture). This way, one end will have a y lbs force and other a force of (100 + 180 + 30) lbs. I don't know if this is possible and I am not totally convinced that the rope is experiencing a force of 100 lbs due to the painter pulling on it.

How can I properly use Newton's third law to determine the impact of the 100 lbs downwards force on the overall system (the painter and bosun's chair)?