
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
Triangle $DEF$ lies in the Cartesian plane and has an area of $84$. The coordinates of $D$ and $E$ are $(10,17)$ and $(25,22),$ respectively, and the coordinates of $F$ are $(x,y).$ The line containing the median to side $DE$ has slope $-3$. Find the largest possible value of $x+y$.

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