
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 that point A $(a, b)$ (where $a \neq b$) is transformed by the matrix $M = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$ to point A' $(-b, a)$:
(I) Find the inverse matrix $M^{-1}$;
(II) Find the equation of the curve $C'$ that is the transformation of curve $C: (x-1)^2 + y^2 = 1$ under the corresponding transformation by matrix $M^{-1}$.

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