
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
Carefully observe the following three rows of related numbers:<br/>First row: $-2$, $4$, $-8$, $16$, $-32$, $64$, $\ldots \ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $66$, $\ldots \ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $32$, $\ldots \ldots$;<br/>Answer the following questions:<br/>$(1)$ The $8$th number in the first row is ______;<br/>$(2)$ The $n$th number in the second row is ______, and the $n$th number in the third row is ______;<br/>$(3)$ Taking the $n$th number from each row, is there a value of $n$ such that the sum of these three numbers is $-1278$? If it exists, find the value of $n$; if it does not exist, please explain the reason.

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