
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 that the vector $$\begin{bmatrix} 1 \\ -1 \end{bmatrix}$$ is an eigenvector of matrix $A$ corresponding to the eigenvalue $-1$. In the Cartesian coordinate system $xOy$, point $P(1,1)$ is transformed into $P'(3,3)$ under the transformation corresponding to matrix $A$. Find the matrix $A$.

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