
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 $MNP$, $MN = 5$, $NP = 7$, and $MP = 8$. The medians from each vertex intersect at the centroid $G$. Let the projections of $G$ onto $NP$, $MP$, and $MN$ be $X$, $Y$, and $Z$, respectively. The altitude from $N$ to side $MP$ is also given as $4$ units. Calculate $GX + GY + GZ$.

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