
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 the maximum constant $k$ such that $\frac{k a b c}{a+b+c} \leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$.

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