
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
Let $a > 0$, $b > 0$. If matrix **A** transforms the circle $C: x^2 + y^2 = 1$ into the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.  
(1) Find the values of $a$ and $b$;  
(2) Find the inverse matrix **A**$^{-1}$ of matrix **A**.

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