
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 $A= \begin{bmatrix} 1 & -1 \\ a & 1\end{bmatrix}$, where $a\in\mathbb{R}$, if the point P (1, 1) is transformed by matrix A to the point P' (0, -3),

(1) Find the value of the real number $a$;
(2) Determine the eigenvalues and eigenvectors of matrix A.

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