
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  $O$  be the centre of the square  $ABCD$ . Let  $P,Q,R$  be respectively on the segments  $OA,OB,OC$  such that  $OP=3,OQ=5,OR=4$ . Suppose  $S$  is on  $OD$  such that  $X=AB\cap PQ,Y=BC\cap QR$  and  $Z=CD\cap RS$  are collinear. Find  $OS$ .

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