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Logically he should have because he has already narrated how Brahma and others called Rama as Vishnu and Narayana and the supreme being at the end of the war.

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Valmiki addressed Rama with the words like Dashrathe, Mahavrata Rama etc. But when Sita went into Patala, Lord Brahma addressed Lord Rama as Vishnu:

I would not have reminded thee of this secret but for the present necessity. Do thou now once more remember thy birth from Vishnu. By nature, Sita is pure, chaste and ever dependent upon thee. And by virtue of her ascetic dependance upon thee she has gone to the region of Nagas. Thou shalt again meet her in heaven (Vaikuntha). Hear, what I communicate unto thee before the assembly.

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And Sage Valmiki was present there when Lord Brahma said this as at the end of the conversation with Lord Brahma, Lord Rama said to Sage Valmiki:

O illustrious sir, the Rishis living in Brahmas region have grown desirous of hearing my future history. So let it be taken up next morning. enter image description here

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