
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 the matrix $M= \begin{pmatrix} a & 1 \\ b & 1 \end{pmatrix}$, one of its eigenvalues $\lambda$ corresponds to the eigenvector $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$.  
(Ⅰ) Find the inverse matrix of $M$;  
(Ⅱ) Find the new equation of the curve $C: x^2+2xy+2y^2=1$ under the transformation corresponding to matrix $M$.

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