Solving Dynamic Constrained Optimization Problems .Please take me out of this doubt

Hi everyone
I have an optimization problem in my DSGE model and I doubt how to solve it. thank you for taking me out of this doubt. Because if I make a mistake, I will face bigger problems in Dynare.I know optimization, but having a dynamic constraint on my optimization problem makes it a bit complicated. Thank you in advance for your kindness
I will not write the expectation operator for simplicity.

My optimization problem in the simplest case is as follows


Which solution do you think is the best?. The third and fourth solutions seem to be more correct than the others. Please take me out of this doubt. Is there a better way to solve this problem?

I am not sure I understand. Why can you not take the derivative with respect to L_t in approach 1? You will simply get that the second constraint is not binding, i.e. \mu_{L,t}=0. But that is expected here, because nothing restricts L_t.

Thank you very much, Professor. According to your guidance, the answer should be this:

Of course, if we simplify the answer obtained from other approaches, we will reach the same result.I realized what I had done wrong