
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
Let positive real numbers \( a, b, c \) satisfy \( abc = 10^{11} \) and \( \lg a \cdot \lg(bc) + \lg b \cdot \lg(ca) + \lg c \cdot \lg(ab) = 40 \). Then, \( \sqrt{\lg^2 a + \lg^2 b + \lg^2 c} = \quad \).

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