
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  $M$  be the intersection of two diagonal,  $AC$  and  $BD$ , of a rhombus  $ABCD$ , where angle  $A<90^\circ$ . Construct  $O$  on segment  $MC$  so that  $OB<OC$  and let  $t=\frac{MA}{MO}$ , provided that  $O \neq M$ . Construct a circle that has  $O$  as centre and goes through  $B$  and  $D$ . Let the intersections between the circle and  $AB$  be  $B$  and  $X$ . Let the intersections between the circle and  $BC$  be  $B$  and  $Y$ . Let the intersections of  $AC$  with  $DX$  and  $DY$  be  $P$  and  $Q$ , respectively. Express  $\frac{OQ}{OP}$  in terms of  $t$ .

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