
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, the equation of circle O is $x^2+y^2=r^2$ ($r>0$), with two points A(4,0) and B(0,4). The moving point P satisfies $\overrightarrow{AP} = \lambda \overrightarrow{AB}$ ($0 \leq \lambda \leq 1$).
(1) Find the equation of the trajectory C of the moving point P;
(2) If for any point P on the trajectory C, there always exists a line l passing through P intersecting circle O at points M and N, and point M is the midpoint of the line segment PN, find the range of values for r.

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