
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 $DEF$, where $DE = 8$, $EF = 6$, and $DF = 10$:

A point $Q$ is chosen at random inside triangle $DEF$. A circle with radius $1$ is inscribed within the triangle. What is the probability that $Q$ is simultaneously closer to $F$ than to $D$ or $E$, and also inside the inscribed circle?

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