
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 the Cartesian coordinate system $xOy$, it is known that $P$ is the common point of the graph of the function $y=2\ln x$ and the circle $M$: $(x-3)^{2}+y^{2}=r^{2}$, and their tangents at point $P$ coincide. If the graph of the quadratic function $y=f(x)$ passes through points $O$, $P$, and $M$, then the maximum value of $y=f(x)$ is ______.

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