
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
Let $D$ be a point in the plane of triangle $ABC$, and assume $G$ is the centroid of triangle $ABC$. Let $P$ be an arbitrary point. Then there exist constants $k$ and $m$ so that
\[ PA^2 + PB^2 + PC^2 = k\cdot PG^2 + m\cdot PD^2 + GA^2 + GB^2 + GC^2. \]
Find $k$ and $m$ when $D$ is the midpoint of the segment $AB$.

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