
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 chord $AB$ of the ellipse $O: \frac{x^2}{12} + \frac{y^2}{6} = 1$ satisfies $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. Let the projection of the origin $O$ onto $AB$ be $M$. The number of points $M$ at a distance of $1$ from the line $x+y=1$ is ______.

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