
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
The coordinate system $X O Y$ is first translated such that the new origin is at the point $O^{\prime}(2, 3)$, and then the axes are rotated by an angle $\varphi = 30^{\circ}$. Determine the new coordinates of point $A$ if the old coordinates in the $X O Y$ system were $A(-1, 5)$. Determine the coordinates of point $B$ in the old $X O Y$ system if in the new coordinate system it is given as $B(4, -2)$.

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