Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

$x\,\,\,\, z = 15$

$x\,\,\,\, y = 12$

$x\,\,\,\, x = 36$

If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x + 10y + z$.

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