
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
Find the smallest positive integer  $k$  such that  $ \underbrace{11\cdots 11}_{k\text{ 1's}}$  is divisible by  $9999$ .

Let  $T = TNYWR$ . Circles  $\omega_1$  and  $\omega_2$  intersect at  $P$  and  $Q$ .  The common external tangent  $\ell$  to the two circles closer to  $Q$  touches  $\omega_1$  and  $\omega_2$  at  $A$  and  $B$  respectively.  Line  $AQ$  intersects  $\omega_2$  at  $X$  while  $BQ$  intersects  $\omega_1$  again at  $Y$ .  Let  $M$  and  $N$  denote the midpoints of  $\overline{AY}$  and  $\overline{BX}$ , also respectively.  If  $AQ=\sqrt{T}$ ,  $BQ=7$ , and  $AB=8$ , then find the length of  $MN$ .

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