
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 triangle with  $BC=30$ ,  $AC=50$ , and  $AB=60$ .  Circle  $\omega_B$  is the circle passing through  $A$  and  $B$  tangent to  $BC$  at  $B$ ;  $\omega_C$  is defined similarly.  Suppose the tangent to  $\odot(ABC)$  at  $A$  intersects  $\omega_B$  and  $\omega_C$  for the second time at  $X$  and  $Y$  respectively.  Compute  $XY$ .

Let  $T = TNYWR$ . For some positive integer  $k$ , a circle is drawn tangent to the coordinate axes such that the lines  $x + y = k^2, x + y = (k+1)^2, \dots, x+y = (k+T)^2$  all pass through it. What is the minimum possible value of  $k$ ?



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