
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 $A= \begin{bmatrix} 3 & a \\ 0 & -1 \end{bmatrix}$, where $a \in \mathbb{R}$, if point $P(2, -3)$ is transformed by matrix $A$ into point $P'(3, 3)$.
(1) Find the value of the real number $a$;
(2) Find the eigenvalues of matrix $A$ and their corresponding eigenvectors.

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