Updating initial conditions for solve block

Hello ,

I have two equations , two unknowns and one freed parameter.

So solve block finds solutions for two unknowns with respect to changing parameter in the equations.

What i want to do is updating initial guesses for solve block after each solution ( taking the previous solution and setting them initial guess )

I know this will not change anything for this system but i need to apply this continuation method to a very complex and nonlinear set of equations which is very sensitive to initial guesses ( I dont have only one solution depending on initial guesses i have another solutions also )

So i want to be sure that i am following a one branch of solutions by changing initial guesses. In the end i should be able to get a graph between freed parameter and solutions.

Regards