
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
Find a costant  $C$ , such that   $$    \frac{S}{ab+bc+ca}\le  C $$   where  $a,b,c$  are the side lengths of an arbitrary triangle, and  $S$  is the area of the triangle. 
(The maximal number of points is given for the best possible constant, with proof.)

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