
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 $\triangle ABC$, point $G$ satisfies $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\overrightarrow{AM}=m\overrightarrow{AB}$ $(m>0)$ and $\overrightarrow{AN}=n\overrightarrow{AC}$ $(n>0)$, then the minimum value of $3m+n$ is ______.

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