자손이 반성하다 --- it's wrong in this case. (it means regret.)
자손이 번성하다 -- it's correct.
자손 means descendant like babies, children, son & daughter, Grand-son&daughter;번성하다 means Flourish
so, it means they hope their descendant get flourished in the future.
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수수께끼 재밌게 보고 있어요~~