
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 triangle $ABC$, sides $AB$, $BC$, and $CA$ have weights $x$, $y$, and $z$, respectively. Let it be known that the sum of the weights on any two sides is equal to the weight on the third side multiplied by a constant $k$. Find the ratio of the weights on side $AB$ to $BC$ to $CA$.
A) 1:1:1
B) k:1:1 
C) 1:k:k 
D) k:k:k

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