Question: Is an analytical solution possible for this optimization problem?

Is an analytic solution possible for this problem: find an expression for maximum of power subject to constraints f=0, g=0, h=0? The feasible range is: T1, T2, R, Rthhot, Rthcold, and Z>0.

I know how to solve it using either FindMaximum or Langrage multiplier if I assign numerical values to T1, T2, R, Rthhot, Rthcold, and Z. I'd like to know if it's possible to find an analytic solution if I leave T1, T2, R, Rthhot, Rthcold and Z as parameters

I tried this but it did not work:

f := (X, Y, Rload, Rth, S) -> -(T1 - X)/Rthhot + (X - Y)/Rth + S^2*X*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

g := (X, Y, Rload, Rth, S) -> (Y - T2)/Rthcold - (X - Y)/Rth - S^2*Y*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

h := (X, Y, Rload, Rth, S) -> S^2*Rth/R - Z;

power := (X, Y, Rload, Rth, S) -> S^2*(X - Y)^2*Rload/(R + Rload)^2;

LagrangeMultipliers(power(X, Y, Rload, Rth, S), [f(X, Y, Rload, Rth, S), g(X, Y, Rload, Rth, S), h(X, Y, Rload, Rth, S)], [X, Y, Rload, Rth, S]);