
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 distance from the point $(x, -x)$ to $(3, -3)$ is given by $\sqrt{(3-x)^2 + (-3+x)^2} = \sqrt{x^2 - 6x + 9 + x^2 -6x +9} = \sqrt{2x^2-12x+18}$ and the distance from the point $(2x, x)$ to $(3, -3)$ is given by $\sqrt{(2x-3)^2 + (x+3)^2} = \sqrt{4x^2-12x+9+x^2+6x+9} = \sqrt{5x^2-6x+18}$. Find the number of integers $x$ such that the distances from $(x, -x)$ or $(2x, x)$ to $(3, -3)$ is less than or equal to $9$.

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