
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
Let the sum of the digits of a four-digit number $N$ (all non-zero) be denoted as $G(N)$, the product of the thousands digit and the hundreds digit of $N$ be denoted as $P(N)$, and the product of the tens digit and the units digit of $N$ be denoted as $Q(N)$. The absolute value of $\frac{G(N)}{P(N)-Q(N)}$ is called the "companion value" of $N$.  
$(1)$ The "companion value" of $4164$ is ______;  
$(2)$ If the sum of the thousands digit and the units digit of $N$ is divisible by $9$, and $G(N)=16$, and the "companion value" of $N$ is $4$, then the smallest value of $N$ that satisfies the conditions is ______.

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