<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 At a certain pond, goose eggs have varying hatch rates: 1/2 of the eggs laid by the geese hatched, 1/3 of the remaining eggs hatched after a month, and the final 1/6 after two months. Of the geese that hatched, 3/4 of those that hatched immediately survived the first month, 2/3 of those that hatched after a month survived the first month, and 1/2 of those that hatched after two months survived the first month. Following the first month, the survival rate for the first year varied: 3/5 of those that hatched immediately and survived the first month died within the first year, 1/4 of those that hatched after a month and survived the first month died, and 1/3 of those that hatched after two months and survived the first month died within the first year. If 150 geese survived the first year and if no more than one goose hatched from each egg, how many goose eggs were laid at the pond? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
375
At a certain pond, goose eggs have varying hatch rates: 1/2 of the eggs laid by the geese hatched, 1/3 of the remaining eggs hatched after a month, and the final 1/6 after two months. Of the geese that hatched, 3/4 of those that hatched immediately survived the first month, 2/3 of those that hatched after a month survived the first month, and 1/2 of those that hatched after two months survived the first month. Following the first month, the survival rate for the first year varied: 3/5 of those that hatched immediately and survived the first month died within the first year, 1/4 of those that hatched after a month and survived the first month died, and 1/3 of those that hatched after two months and survived the first month died within the first year. If 150 geese survived the first year and if no more than one goose hatched from each egg, how many goose eggs were laid at the pond?
test