
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
Define: For three given numbers in fixed positions: $a$, $b$, $c$, calculate $a-b$, $\frac{a-c}{2}$, $\frac{b-c}{3}$, and consider the minimum value among these three numbers as the "white horse number" of $a$, $b$, $c$. For example, for $1$, $-2$, $3$, since $1-\left(-2\right)=3$, $\frac{1-3}{2}=-1$, $\frac{-2-3}{3}=-\frac{5}{3}$, the "white horse number" of $1$, $-2$, $3$ is $-\frac{5}{3}$.<br/>$(1)$ The "white horse number" of $-2$, $-4$, $1$ is ______;<br/>$(2)$ By rearranging the three numbers "$-2$, $-4$, $1$", different "white horse numbers" are obtained. Find the maximum value among these different "white horse numbers" ______;<br/>$(3)$ By rearranging $-1$, $6$, $x$, different "white horse numbers" are obtained. If one of the "white horse numbers" is $2$, find the value of $x$.

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