Function range with numerical method

In[1]  FunctionRange[{{r^(1/2) Cos[t/2] + r^(1/3) Cos[t/3],             r^(1/2) Sin[t/2] + r^(1/3) Sin[t/3]},            0 <= r <= 1 && 0 <= t <= Pi/2}, {r, t}, {x, y}]  

Gives

Out[1]  0. <= x <= 2. && 0. <= y <= 1.20711 

with an error message :

enter image description here

FunctionRange: Unable to find the exact range. Returning bounds on the range computed using numeric optimization methods. 

Loos like mathematica attacked this problem algebraically at first, and failed, and tried numerical method. My question is : is it possible to attack this problem numerically from the beginning ?

My guess is

In[1]  FunctionRange[{{r^(1/2) Cos[t/2] + r^(1/3) Cos[t/3],             r^(1/2) Sin[t/2] + r^(1/3) Sin[t/3]},            0 <= r <= 1 && 0 <= t <= Pi/2}, {r, t}, {x, y}, Method -> blahblah] 

But I don’t know what blahblah is.