
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
In rectangle $ABCD$ with sides $AB = 2$ and $BC = 1$, a circle $\omega$ is inscribed touching all sides. It intersects $\overline{CD}$ at $M$, and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is the length of segment $AP$?

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