
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
Two separate sets of side lengths for triangles are given: 
1) $(x, x\sqrt{3}, 2x)$, where $x=15$ and 
2) $(3x, 4x, 5x)$ with $x=13$. 
Verify if both these sets represent right triangles. If they do, calculate the angle $\theta$ between sides $15$ and $15\sqrt{3}$ in the first triangle using the Law of Cosines, and establish whether the hypotenuse of the second triangle can be considered as the diameter of a circle.
A) 30
B) 65
C) 75
D) 130

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