
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
Point $I$ is the center of the inscribed circle of triangle $ABC$, $M$ is the midpoint of the side $AC$, and $W$ is the midpoint of the arc $AB$ of the circumcircle, not containing $C$. It turns out that $\angle AIM = 90^\circ$. In what ratio does point $I$ divide the segment $CW$?

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