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Does the divergent solution of this equation :$f'=e^{f^{-1}}$ of Gevrey type and could be Borel summation applied for it?

This question was asked here in MO by someone seeking for the solution of the functional -differential:$f'=e^{f^{-1}}$ not exactly an O.D.E, and again here seeking for the growth rate of it solution ...
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