
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
Three people, \( A \), \( B \), and \( C \), visit their teacher \( D \)'s house. Teacher \( D \) gives each person a hat with a four-digit number written on it. It is known that these three four-digit numbers are perfect squares (for example, 4 and 100 are perfect squares as they are squares of 2 and 10 respectively). Additionally, these four-digit numbers have a tens digit of 0, and their units digit is not 0. Each person can only see the numbers on the hats of the other two individuals. These three individuals are very clever and honest, and the following conversation occurs:
\( A \) says, “The units digit of the numbers on \( B \) and \( C \)'s hats are the same.”
\( B \) and \( C \) simultaneously say, “After hearing \( A \)'s statement, I know what my number is.”
\( A \) then says, “After hearing \( B \) and \( C \)'s statements, I also know what my number is. The units digit of my number is an even number.”
Find the sum of the numbers on the hats of \( A \), \( B \), and \( C \).

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