
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $5$, and suppose we have ${AB_5}+{C_5}={C}0_5$ and ${AB_5}+{BA_5}={CC_5}$. Find the three-digit number ${ABC}$. (Here, $AB_5$ represents a base-5 number with digits $A$ and $B$, and similarly for the other numbers).

Present the answer in LaTex format: \boxed{Your answer}
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