How can I restore a vintage veneer bedroom set?

L
by L
I have my parents 1950s bedroom set. The veneer is chipped and small pieces are missing in a few places. Can I stain the base where the pieces are missing by matching color as closely as possible? How can I polish/ restore the luster to the furniture? Any product and method suggestions would be appreciated. I don’t want to replace the veneer. Thank you!

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