
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: $-2$, $4$, $-8$, $16$, $-32$, $\ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $\ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $\ldots$;<br/>Answer the following questions:<br/>$(1)$ The $6$th number in the first row is ______;<br/>$(2)$ What is the relationship between the numbers in the second row, the third row, and the first row?<br/>$(3)$ Take a number $a$ from the first row and the other two numbers corresponding to it from the second and third rows, such that the sum of these three numbers is $642$. Find the value of $a$ and state which number in the first row $a$ corresponds to.

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