Put value in Option-embedded Bonds when rates go up

We know that when rates go UP , call prices go UP and put prices go DOWN.

Why is it then when rates go UP , the put value goes UP in an option-embedded bond (Even though we are long the put option)?

because when interest rates go up, investor has the option to exercise the put and receive a bond that pays higher interest to this investor


I am not so sure about the first phrase, as far as I know when rates go up, this means that bond prices will go down and the put will be in the money accordingly the put price will go up. I do not know if that will clarify your question.