2 added 212 characters in body
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05AB1E, 7 bytes

LOΔSPSO

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Exlpanation

L         # push range [1 ... input]
 O        # sum range
  Δ       # loop until top of stack stops changing
   SP     # product of digits
     SO   # sum of digits

05AB1E, 7 bytes

LOΔSPSO

Try it online!

05AB1E, 7 bytes

LOΔSPSO

Try it online!

Exlpanation

L         # push range [1 ... input]
 O        # sum range
  Δ       # loop until top of stack stops changing
   SP     # product of digits
     SO   # sum of digits
1
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05AB1E, 7 bytes

LOΔSPSO

Try it online!