
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
Two circles  $\omega_1$  and  $\omega_2$  are said to be  $\textit{orthogonal}$  if they intersect each other at right angles.  In other words, for any point  $P$  lying on both  $\omega_1$  and  $\omega_2$ , if  $\ell_1$  is the line tangent to  $\omega_1$  at  $P$  and  $\ell_2$  is the line tangent to  $\omega_2$  at  $P$ , then  $\ell_1\perp \ell_2$ .  (Two circles which do not intersect are not orthogonal.)

Let  $\triangle ABC$  be a triangle with area  $20$ .  Orthogonal circles  $\omega_B$  and  $\omega_C$  are drawn with  $\omega_B$  centered at  $B$  and  $\omega_C$  centered at  $C$ .  Points  $T_B$  and  $T_C$  are placed on  $\omega_B$  and  $\omega_C$  respectively such that  $AT_B$  is tangent to  $\omega_B$  and  $AT_C$  is tangent to  $\omega_C$ .  If  $AT_B = 7$  and  $AT_C = 11$ , what is  $\tan\angle BAC$ ?

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