
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
Observe the following three rows of numbers:  
$-3$, $9$, $-27$, $81$, $\ldots$ ①  
$6$, $-18$, $54$, $-162$, $\ldots$ ②  
$-1$, $11$, $-25$, $83$, $\ldots$ ③  
$(1)$ The $n$th number in the first row is _______; the $n$th number in the second row is _______; the $n$th number in the third row is _______; (express using $n$)  
$(2)$ In the second row, there exist three consecutive numbers whose sum is $378$, these three numbers are ______, ______, ______;  
$(3)$ Let $x$, $y$, $z$ be the 2024th number in each row, find the value of $x+y+z$.

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