
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 matrix $A= \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$.  
① Find the inverse matrix $A^{-1}$.  
② If a curve $C$ is transformed into curve $C'$ under the action of matrix $A^{-1}D$, and $C'$ is defined by the equation $x^2 - y^2 = 1$, find the equation of curve $C$.

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