
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 $DEF$, $DE = DF = 120$, and $EF = 72$. Circle $R$ has radius $20$ and is tangent to $\overline{DF}$ and $\overline{EF}$. Circle $S$ is externally tangent to $R$ and is tangent to $\overline{DE}$ and $\overline{EF}$. No point of circle $S$ lies outside of $\triangle DEF$. The radius of circle $S$ can be expressed in the form $p - q\sqrt {u}$, where $p$, $q$, and $u$ are positive integers and $u$ is the product of distinct primes. Find $p + qu$.

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