
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
The rectangular spiral shown in the diagram is constructed as follows. Starting at \((0,0)\), line segments of lengths \(1,1,2,2,3,3,4,4, \ldots\) are drawn in a clockwise manner. The integers from 1 to 1000 are placed, in increasing order, wherever the spiral passes through a point with integer coordinates (that is, 1 at \((0,0)\), 2 at \((1,0)\), 3 at \((1,-1)\), and so on). What is the sum of all of the positive integers from 1 to 1000 which are written at points on the line \(y=-x\)?

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