
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
Read the following material:<br/>When studying a mathematical problem, Xiaoding encountered a definition: for three ordered numbers $x_{1}$, $x_{2}$, $x_{3}$, they are called a sequence $x_{1}$, $x_{2}$, $x_{3}$. Calculate $|x_{1}|$, $\frac{|{x}_{1}+{x}_{2}|}{2}$, $\frac{|{x}_{1}+{x}_{2}+{x}_{3}|}{3}$, and the minimum value of these three numbers is called the value of the sequence $x_{1}$, $x_{2}$, $x_{3}$. For example, for the sequence $2$, $-1$, $3$, because $|2|=2$, $\frac{|2+(-1)|}{2}=\frac{1}{2}$, $\frac{|2+(-1)+3|}{3}=\frac{4}{3}$, so the value of the sequence $2$, $-1$, $3$ is $\frac{1}{2}$. Xiaoding further discovered that when changing the order of these three numbers, the obtained sequences can all be calculated for their respective values using the above method. For example, the value of the sequence $-1$, $2$, $3$ is $\frac{1}{2}$; the value of the sequence $3$, $-1$, $2$ is $1$; ... After studying, Xiaoding found that for the numbers "$2$, $-1$, $3$", the minimum value of the values obtained by different permutations is $\frac{1}{2}$. Based on the above material, answer the following questions:<br/>$(1)$ The value of the sequence $-4$, $-3$, $2$ is ______;<br/>$(2)$ By arranging the numbers "$-4$, $-3$, $2$" in different orders, several sequences can be obtained, and the minimum value of the values of these sequences is ______, and the sequence that achieves the minimum value is ______ (write one is enough);<br/>$(3)$ By arranging the numbers $2$, $-9$, $a\left(a \gt 1\right)$ in different orders, several sequences can be obtained. If the minimum value of the values of these sequences is $1$, then the value of $a$ is ______.

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