
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 $\triangle ABC$, the lines connecting each vertex to the midpoint of the opposite side intersect at a point, defined as the centroid $G$ of the triangle. It is easy to see that $\overrightarrow{AG}=\frac{1}{3}(\overrightarrow{AB}+\overrightarrow{AC})$. Similarly, in the tetrahedron $P-ABC$, the lines connecting each vertex to the centroid of the opposite face intersect at a point, defined as the centroid $G$ of the tetrahedron. If $\overrightarrow{PA}=\overrightarrow{a}$, $\overrightarrow{PB}=\overrightarrow{b}$, $\overrightarrow{PC}=\overrightarrow{c}$, then $\overrightarrow{PG}=\_\_\_\_\_\_$ (expressed in terms of $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$).

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