
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 $\overline{CF}$ be an altitude of $\triangle DEF$. Let $P\,$ and $Q\,$ be the points where the circles inscribed in the triangles $DFE\,$ and $EFC$ are tangent to $\overline{CF}$. If $DE = 996\,$, $DF = 995\,$, and $EF = 994\,$, then $PQ\,$ can be expressed as $m/n\,$, where $m\,$ and $n\,$ are relatively prime integers. Find $m + n\,$.

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