
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 polar coordinate system, where O is the pole, it is known that the center of circle C is at $$(1, \frac{\pi}{4})$$, and its radius $r=1$. Point P moves on circle C.  
(Ⅰ) Find the polar equation of circle C;  
(Ⅱ) In the Cartesian coordinate system (using the same unit length as the polar coordinate system, and taking the pole O as the origin, with the polar axis as the positive x-axis), if Q is the midpoint of the line segment OP, find the Cartesian equation for the trajectory of point Q.

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