In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$?

$$ \begin{array}{cccc}
 & a & b & c \\
 & d & e & f \\
+ & g & h & i \\
\hline
1 & 6 & 6 & 5 \\
\end{array} $$

Let's think step by step and output the final answer within \boxed{}.
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