
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
In regular hexagon $ABCDEF$, points $P$ and $Q$ are the midpoints of sides $\overline{BC}$ and $\overline{EF}$ respectively. Find the ratio of areas $[ABPQ]/[CDPQ]$. [asy]
pair A,B,C,D,E,F;
A=(0,2); B=(sqrt(3),1); C=(sqrt(3),-1); D=(0,-2); E=(-sqrt(3),-1); F=(-sqrt(3),1);
draw(A--B--C--D--E--F--cycle);
pair P=(B+C)/2; pair Q=(E+F)/2;
draw(P--Q);

label("$A$",A,N); label("$B$",B,NE); label("$C$",C,E); 
label("$D$",D,S); label("$E$",E,SW); label("$F$",F,W);
label("$P$",P,NE); label("$Q$",Q,W);
[/asy]

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