Question on Eliminate

Questions
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1

In the code below, Eliminate is called to eliminate one of 4 variables appearing in 2 equations. I would think that the result would be 1 equation in the 3 remaining variables. Instead it produces 8 equations, none of which eliminate the target variable. Clearly there is something here I don’t understand. ???

In[1]:= eqs = {x == (7 + Cos[u/3 - 2 v] + 2 Cos[u/3 + v]) Sin[u], 
   y == Cos[u] (7 + Cos[u/3 - 2 v] + 2 Cos[u/3 + v])};

In[2]:= Eliminate[eqs, u]

During evaluation of In[2]:= Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information. >>

Out[2]= 3 (u/3 - 2 v) == u - 6 v && 
 3 (u/3 + v) == u + 3 v && (x^2 + y^2) Cos[u]^2 == y^2 && 
 2 y Cos[u/3 + v] == -7 y + x^2 Cos[u] + y^2 Cos[u] - 
   y Cos[u/3 - 2 v] && 
 2 Cos[u] Cos[u/3 + v] == 
  y - 7 Cos[u] - 
   Cos[u] Cos[u/3 - 2 v] && (28 + 4 Cos[u/3 - 2 v]) Cos[u/3 + v] + 
   4 Cos[u/3 + v]^2 == -49 + x^2 + y^2 - 14 Cos[u/3 - 2 v] - 
   Cos[u/3 - 2 v]^2 && Sin[u/3 - 2 v]^2 == 1 - Cos[u/3 - 2 v]^2 && 
 4 Sin[u/3 + v]^2 == 
  53 - x^2 - y^2 + 14 Cos[u/3 - 2 v] + Cos[u/3 - 2 v]^2 + 
   28 Cos[u/3 + v] + 4 Cos[u/3 - 2 v] Cos[u/3 + v]
2

I tried to do it with Reduce, but it’s not returning an answer in any reasonable amount of time.