সহায়িকা

Explanation
The phrase "would rather" is used to express preference between two actions, using the structure "would rather [verb] than [verb]".
In this sentence, the boy is expressing a clear preference for starving over begging.
The complete, grammatically correct sentence is: "I would rather starve than beg". The word "then" in the original sentence is likely a typo for "than".
Why other options are incorrect
better and would better are incorrect because "had better" or "would better" are used for giving advice or recommendations, not for expressing preference using the "than" structure.
rather is incorrect on its own as it needs the auxiliary verb "would" (or "had") to form the correct idiomatic expression for indicating preference in this comparative context

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