
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 point $P(a, b)$, it is first transformed by the matrix $M = \begin{bmatrix} \dfrac {1}{2} & -\dfrac { \sqrt {3}}{2}\\ \dfrac { \sqrt {3}}{2} & \dfrac {1}{2} \end{bmatrix}$ and then by the matrix $N = \begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}$. The coordinates of the resulting point are $(8, 4\sqrt{3})$. Find the values of the real numbers $a$ and $b$.

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