This does solve the problem for me:

LLMSynthesize["A mountain climber is on a mountain whose shape is
given by the graph of the function f(x,y)=10-2x^2-y^2 at point 
P=(1,2,4). Find a track gamma(t) contained in the function that 
maintains gamma(0)=P, that if our climber climbs along it then at any 
point at this track the direction of his movement will be in the 
direction of the steepest ascent.Note that the direction of steepest 
decent will change over time. Do not use the steepest ascent only at 
the starting point. Consider that it changes. Solve the resulting 
ODEs.", LLMEvaluator -> LLMConfiguration["Model" -> "gpt-4"]]

Here is a visualisation of the result (also generated by GPT):

Cheers,

Marco