<|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
 In the Cartesian coordinate plane $(xOy)$, point $A$ is on the ellipse $\frac{x^{2}}{25}+ \frac{y^{2}}{9}=1$, and point $P$ satisfies $\overrightarrow{AP}=(λ-1) \overrightarrow{OA}(λ∈R)$, and $\overrightarrow{OA}⋅ \overrightarrow{OP}=72$. Find the maximum length of the projection of line segment $OP$ on the $x$-axis. 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
15
In the Cartesian coordinate plane $(xOy)$, point $A$ is on the ellipse $\frac{x^{2}}{25}+ \frac{y^{2}}{9}=1$, and point $P$ satisfies $\overrightarrow{AP}=(λ-1) \overrightarrow{OA}(λ∈R)$, and $\overrightarrow{OA}⋅ \overrightarrow{OP}=72$. Find the maximum length of the projection of line segment $OP$ on the $x$-axis.
test