
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
Given the four-digit number $\overline{ABCD}$, it satisfies the following properties: $\overline{AB}$, $\overline{BC}$, and $\overline{CD}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer. For example, $4=2^{2}, 81=9^{2}$, so we call $4$ and $81$ perfect squares). Find the sum of all four-digit numbers that satisfy these properties.

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