
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 for any plane vector $\overrightarrow{AB}=(x,y)$, rotating the vector $\overrightarrow{AB}$ counterclockwise around its starting point by an angle $\theta$ results in the vector $\overrightarrow{AP}=(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta)$, which is called rotating point $B$ counterclockwise around point $A$ by an angle $\theta$ to get point $P$. If point $A(1,2)$ and point $B(1+\sqrt{3},4)$ are given, and point $B$ is rotated clockwise around point $A$ by $\frac{π}{3}$ to get point $P$, find the coordinates of point $P$.

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