
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
Let  $ABC$  be a scalene triangle with inradius  $1$  and exradii  $r_A$ ,  $r_B$ , and  $r_C$  such that \[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\] If \[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\] then the area of  $\triangle{ABC}$  can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

*Proposed by Tristan Shin*

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