
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
If a three-digit number $m$, where the hundreds digit is the sum of the units digit and the tens digit, is called a "unity number". Move the hundreds digit of $m$ to the units digit to form a new three-digit number $n$, define $p = m - n$, $f(m) = \frac{p}{9}$. For example, $981$ is a "unity number". Moving $9$ to the units digit, we get a new three-digit number $n = 819$, $p = 981 - 819 = 162$, then $f(m) = \frac{{162}}{9} = 18$.  
$(1)$ The minimum value of $f(m)$ is ______, find the corresponding "unity number" $m$ at this time;  
$(2)$ If $f(m)$ is a multiple of $12$, then $f(m)$ is called a "zodiac number". Find all "zodiac numbers".

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