
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 $A=\begin{vmatrix} a & 1 \\ 0 & b \end{vmatrix}$ transforms the point $(1,1)$ into the point $(2,2)$, find the values of $a$ and $b$. Also, find the equation of the curve $C: x^2+y^2=1$ under the transformation of matrix $A$.

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