
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 columns of numbers:<br/>$-1$, $+3$, $-5$, $+7$, $-9$, $+11$, $\ldots$ ①<br/>$-3$, $+1$, $-7$, $+5$, $-11$, $+9$, $\ldots$ ②<br/>$+3$, $-9$, $+15$, $-21$, $+27$, $-33$, $\ldots$ ③<br/>$(1)$ The $10$th number in column ① is ______, and the $15$th number in column ② is ______;<br/>$(2)$ In column ②, are there three consecutive numbers whose sum is $1001$? If yes, find these three numbers; if not, explain why;<br/>$(3)$ If we take the $k$th number in each column, and the sum of these three numbers is exactly $599$, find the value of $k$.

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