
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 $a, b$ are real numbers, the transformation $T$ corresponding to matrix $M= \begin{bmatrix} a & - \frac {1}{2} \\ \frac {1}{2} & b\end{bmatrix}$ transforms the point $(2, 2)$ into point $P′(\sqrt {3}-1, \sqrt {3}+1)$.  
(1) Find the values of the real numbers $a, b$;  
(2) Find the inverse matrix $N$ of matrix $M$.

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