
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
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 "sum multiple" of $m". For example, since $247\div \left(2+4+7\right)=247\div 13=19$, therefore $247$ is a "sum multiple" of $13". Another example is: since $214\div \left(2+1+4\right)=214\div 7=30\ldots \ldots 4$, therefore $214$ is not a "sum multiple". $(1)$ Determine whether $357$ and $441$ are "sum multiples". Explain the reasons. $(2)$ A three-digit number $A$ is a "sum multiple" of $12$, where $a$, $b$, $c$ are the digits of number $A$ and $a \gt b \gt c$. Choose any two digits from $a$, $b$, $c$ to form a two-digit number, with the largest two-digit number denoted as $F(A)$ and the smallest two-digit number denoted as $G(A)$. If $\frac{{F(A)+G(A)}}{{16}}$ is an integer, find all numbers $A$ that satisfy the condition.

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