Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $6$, and suppose we have ${AB_6}+{C_6}={C}0_6$ and ${AB_6}+{BA_6}={CC_6}$. Find the three-digit number ${ABC}$. (Interpret $AB_6$ as a base-6 number with digits $A$ and $B$, not as $A$ times $B$. The other expressions should be interpreted in this way as well).
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