
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
In a circle with a radius of $6$ units, $PQ$ and $RS$ are perpendicular diameters. A chord $PT$ intersects diameter $RS$ at point $V$, and the length of chord $PT$ is $10$ units. Find the lengths of the two segments into which the diameter $RS$ is divided by point $V$.
**A)** $6-\sqrt{11}, 6+\sqrt{11}$  
**B)** $5-\sqrt{6}, 5+\sqrt{6}$  
**C)** $4-\sqrt{5}, 4+\sqrt{5}$  
**D)** $6, 6$  
**E)** $7-\sqrt{11}, 7+\sqrt{11}$

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