
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 value of the expression

$$
27abc + a\sqrt{a^2 + 2bc} + b\sqrt{b^2 + 2ca} + c\sqrt{c^2 + 2ab}
$$

where \(a, b, c\) are positive real numbers such that \(a + b + c = \frac{1}{\sqrt{3}}\).

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