
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 the coordinates of the vertices of triangle $OAB$ as $O(0,0)$, $A(2,9)$, $B(6,-3)$, and point $P$ with an x-coordinate of $14$ and $\overrightarrow{OP}=\lambda \overrightarrow{PB}$. Point $Q$ lies on edge $AB$ and satisfies $\overrightarrow{OP}\bullet \overrightarrow{AP}=0$.

(1) Find the value of the real number $\lambda$ and the coordinates of point $P$.
(2) Find the coordinates of point $Q$.
(3) If $R$ is a moving point on line segment $OQ$ (including the endpoints), find the range of values for $\overrightarrow{RO} \bullet (\overrightarrow{RA}+\overrightarrow{RB})$.

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