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I'm confused by who is doing what.

Where does the program listen for and know about your clicks? How does that relate to the Robot class's clicks?
What program updates the screen?

What screen?...

Ok, which step of the logic are you having trouble coding?

That sounds like you need to work out the logic for all the different cases and program them using if statements etc.
There are many techniques for working out logic. I use flowcharts. Some use...

What part of the problem is not working?
Can you find the pixel with the color?
Can you move the mouse to that location?

If it can't find the pixel on the first pass over the image, what changes...

Please explain what the problem is. What does that statement mean?
What is the "check" and what is the "do as the methods"?

Which of the listed steps does the program do correctly and which...