What do you do when you are certain that all the possible ways of achieving something are exhausted?

I’ve been there and all you can do is count on the fact that things will work out one way or another. They usually do even if it is not your preferred choice.
It also depends on what you are trying to achieve and if your expectations are too high. Big difference in exhausting all possibilities if you’re teetering on the verge of something dire like homelessness vs. not being able to save a million dollars in 2 years. lol

Yes, there comes a time when you have to leave things to luck, chance, and just trust that somehow, some way, it will all come together.

Stay open to the unexpected.

That could be an unexpected shift in circumstances that opens up a way forward.

It could be an epiphany, a solution so creative that it can only surface when the obvious has been eliminated.

And it could also be the unexpected discovery that you have changed, and what seemed so important before no longer is.

It depends on the circumstances. What is ‘it’? If it’s a troubled relationship, in time I will accept it’s run it’s course and move on. If it’s a project I might put it on the back-burner until another solution emerges.

I tend to be in the ‘never give up’ camp, but I’ve learned from experience that giving up can sometimes be exactly the right thing to do.

If possible, you move on. The agony of course is in those situations where either circumstance or disposition impede escape.

I have tried to live by the maxim:

If at first you don’t succeed, try again.
If you don’t succeed the second time, give up
No use making a damned fool of yourself.