
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 line segment AB of length 2 with midpoint C. When the endpoints A and B of line segment AB move on the x-axis and y-axis respectively, the trajectory of point C is curve C<sub>1</sub>;  
(1) Find the equation of curve C<sub>1</sub>;  
(2) The line $\sqrt {2}ax+by=1$ intersects curve C<sub>1</sub> at points C and D (where $a$, $b$ are real numbers), and $\triangle COD$ is a right-angled triangle (O is the origin). Find the minimum distance between point P($a$, $b$) and point (0, 1).

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