
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 second-order matrix $$M= \begin{pmatrix} \left.\begin{matrix}2 & 1 \\ a & b\end{matrix}\right.\end{pmatrix}$$ (where $a, b \in \mathbb{R}$), if matrix $M$ has an eigenvector $$\overrightarrow {α_{1}}$$ corresponding to the eigenvalue -1, which is $$\begin{pmatrix} \left.\begin{matrix}-1 \\ 3\end{matrix}\right.\end{pmatrix}$$, and an eigenvector $$\overrightarrow {α_{2}}$$ corresponding to the eigenvalue 3, which is $$\begin{pmatrix} \left.\begin{matrix}1 \\ 1\end{matrix}\right.\end{pmatrix}$$．  
(I) Find the values of the real numbers $a$ and $b$;  
(II) If vector $$\overrightarrow {β}$$ is $$\begin{pmatrix} \left.\begin{matrix}-3 \\ 5\end{matrix}\right.\end{pmatrix}$$, calculate the value of $M^{5} \overrightarrow {β}$．

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