
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
Given a circle with radius $r$ and center $O$, and a fixed point $P$ inside the circle such that $OP = a$. Consider a randomly chosen point $M$ inside the circle and denote $M^{\prime}$ as the symmetric point of $M$ with respect to $OP$. What is the probability that the circle passing through the points $M$, $P$, and $M^{\prime}$ lies entirely inside the circle with center $O$ and also contains the point $O$? For example, $a = 0.5r$.

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