
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
In triangle \( \triangle ABC \), \( AB = \sqrt{2} \), \( AC = \sqrt{3} \), and \( \angle BAC = 30^\circ \). Let \( P \) be an arbitrary point in the plane containing \( \triangle ABC \). Find the minimum value of \( \mu = \overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA} \).

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