
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 radius $7$, two chords $PQ$ and $RS$ intersect at point $T$ such that $T$ bisects $PQ$ and is $3$ units away from the center of the circle. If $PT=8$, find the length of $RS$ such that the product of the lengths of segments into which $T$ divides $RS$ is a perfect square.

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