
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
Right triangle $T$ is inscribed in circle $A$, which has radius $13$. Circle $B$ with radius $4$ is internally tangent to circle $A$ at one vertex of $T$. Circle $C$, with radius $3$, is internally tangent to circle $A$ at another vertex of $T$. The circles $B$ and $C$ are externally tangent to another circle $E$, which has radius $r$. If the angle between the radii of circle $A$ touching vertices related to circles $B$ and $C$ is $90^\circ$, find $r$.

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