
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
Determine the maximum possible real value of the number  $ k$ , such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers  $ a,b,c\ge 0$  with  $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$ .

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