
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]


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For a three-digit natural number $N$ where each digit is not $0$, if $N$ can be divided by the sum of its digits $m$, then $N$ is called a "whole multiple" of $m". For example: Since $135\div \left(1+3+5\right)=135\div 9=15$, therefore $135$ is a "whole multiple" of $9$. Another example is $524\div \left(5+2+4\right)=524\div 11=47\cdots \cdots 7$, therefore $524$ is not a "whole multiple" of $11$. Let $A$ be a three-digit number that is a "whole multiple" of $12$, where $a$, $b$, and $c$ are the digits of number $A$ and $c \lt b \lt a$. Choose any two digits from $a$, $b$, and $c$ to form a two-digit number, with the larger two-digit number denoted as $P\left(A\right)$ and the smaller two-digit number denoted as $Q\left(A\right)$. If $\frac{P(A)+Q(A)}{2}$ is an integer, find the smallest value of the number $A$ that satisfies this condition.

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