
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 a point  $ M$  found in the same plane with the triangle  $ ABC$ , but not found on any of the lines  $ AB,BC$  and  $ CA$ . Denote by  $ S_1,S_2$  and  $ S_3$  the areas of the triangles  $ AMB,BMC$  and  $ CMA$ , respectively. Find the locus of  $ M$  satisfying the relation:
 $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$ 

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