
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 point $G$ is the centroid of $\triangle ABO$, and $M$ is the midpoint of side $AB$. Then:

$(1)$ Find $\overrightarrow{GA}+ \overrightarrow{GB}+ \overrightarrow{GO} =$\_\_\_\_\_\_\_\_;

$(2)$ If line $PQ$ passes through the centroid $G$ of $\triangle ABO$, and $\overrightarrow{OA} =a$, $\overrightarrow{OB} =b$, $\overrightarrow{OP} =ma$, $\overrightarrow{OQ} =nb$, find: $\frac{1}{m}+ \frac{1}{n} =$\_\_\_\_\_\_\_\_.

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