
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 $7$, and suppose we have ${AB_7} + {C_7} = {C}0_7$ and ${AB_7} + {BA_7} = {CC_7}$. Find the three-digit number ${ABC}$. (Interpret $AB_7$ as a base-7 number with digits $A$ and $B$, not as $A$ times $B$. The other expressions should be interpreted in this way as well).

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