
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 $DE=10, EF=24,$ and $FD=26.$ Circle $\omega_3$ passes through $E$ and is tangent to line $DF$ at $D.$ Circle $\omega_4$ passes through $F$ and is tangent to line $DE$ at $D.$ Let $M$ be the intersection of circles $\omega_3$ and $\omega_4$ not equal to $D.$ Compute the length of segment $MD.$

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