
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  $A_0BC_0D$  be a convex quadrilateral inscribed in a circle  $\omega$ . For all integers  $i\ge0$ , let  $P_i$  be the intersection of lines  $A_iB$  and  $C_iD$ , let  $Q_i$  be the intersection of lines  $A_iD$  and  $BC_i$ , let  $M_i$  be the midpoint of segment  $P_iQ_i$ , and let lines  $M_iA_i$  and  $M_iC_i$  intersect  $\omega$  again at  $A_{i+1}$  and  $C_{i+1}$ , respectively. The circumcircles of  $\triangle A_3M_3C_3$  and  $\triangle A_4M_4C_4$  intersect at two points  $U$  and  $V$ . 

If  $A_0B=3$ ,  $BC_0=4$ ,  $C_0D=6$ ,  $DA_0=7$ , then  $UV$  can be expressed in the form  $\tfrac{a\sqrt b}c$  for positive integers  $a$ ,  $b$ ,  $c$  such that  $\gcd(a,c)=1$  and  $b$  is squarefree. Compute  $100a+10b+c $ .

*Proposed by Eric Shen*

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