<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Given an equilateral triangle $\triangle ABC$ with side length $2\sqrt{3}$, $P$ is a point in the plane of the triangle, and $|\overrightarrow{AP}-\overrightarrow{AB}-\overrightarrow{AC}|=1$. Find the maximum value of $|\overrightarrow{AP}|$. Express your answer as a numerical value. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
7
Given an equilateral triangle $\triangle ABC$ with side length $2\sqrt{3}$, $P$ is a point in the plane of the triangle, and $|\overrightarrow{AP}-\overrightarrow{AB}-\overrightarrow{AC}|=1$. Find the maximum value of $|\overrightarrow{AP}|$. Express your answer as a numerical value.
test