
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 $DEF$ has side lengths $DE=6, EF=10,$ and $FD=8.$ Circle $\omega_1$ passes through $E$ and is tangent to line $FD$ at $D.$ Circle $\omega_2$ passes through $F$ and is tangent to line $DE$ at $D.$ Let $L$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $D.$ Then $DL=\tfrac pq,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

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