
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
If the circle $C: x^2 + y^2 = 1$ is transformed by the matrix $A= \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ ($a>0$, $b>0$) into the ellipse $E: \frac{x^2}{4} + \frac{y^2}{3} = 1$.

(Ⅰ) Find the values of $a$ and $b$;  
(Ⅱ) Determine whether the matrix $A$ is invertible. If it is invertible, find the inverse matrix $A^{-1}$. If it is not invertible, explain why.

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