
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 $\triangle ABC$, \(M\) and \(N\) are points on segments \(AB\) and \(AC\) respectively, with \(AM = \frac{1}{2} AB\) and \(AN = \frac{1}{3} AC\). Points \(E\), \(F\), and \(G\) are points on segment \(BC\). Given that \(\overrightarrow{AE} + \overrightarrow{AF} + \overrightarrow{AG} = x \overrightarrow{AM} + y \overrightarrow{AN}\) for \(x, y \in \mathbf{R}\), find the minimum value of \(\frac{2}{x} + \frac{3}{y}\).

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