
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 a matrix $A = \begin{bmatrix} a & 1 \\ b & 0\end{bmatrix}$, where $a, b \in \mathbb{R}$, when point P$(1,1)$ is transformed by matrix $A$ to point $Q(3,3)$, and given the vector $\vec{\beta} = \begin{bmatrix} 5 \\ 9\end{bmatrix}$.  
(1) Find the values of $a$ and $b$ as well as the eigenvalues and eigenvectors of matrix $A$;  
(2) Calculate $A^{20} \vec{\beta}$.

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