
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
Consider equilateral triangle  $ABC$  with side length  $1$ . Suppose that a point  $P$  in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant  $\kappa$ . Compute the sum of all possible values of  $\kappa$ .

*2018 CCA Math Bonanza Lightning Round #3.4*

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