
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
Given that point \( G \) is the centroid of triangle \( \triangle ABC \), and \( \overrightarrow{AB} = \mathbf{a} \), \( \overrightarrow{AC} = \mathbf{b} \). If line \( PQ \) passes through \( G \) and intersects \( AB \) and \( AC \) at points \( P \) and \( Q \) respectively, with \( \overrightarrow{AP} = m \mathbf{a} \) and \( \overrightarrow{AQ} = n \mathbf{b} \), find the value of \( \frac{1}{m} + \frac{1}{n} \).

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