
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

The decimal representation of the four-digit number ABCD, containing no nines, was divided into three parts in three different ways, resulting in three quadratic equations $Ax^{2} + Bx + CD = 0$, $Ax^{2} + BCx + D = 0$, and AB $x^{2} + Cx + D = 0$. It turns out that they all have roots. Find all possible values of ABCD.

Here AB, BC, and CD are not products of the digits but two-digit numbers. Additionally, in the decimal representation of the number ABCD and the two-digit numbers, the leading digits cannot be equal to 0.

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