
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
Equilateral triangles  $ ABQ$ ,  $ BCR$ ,  $ CDS$ ,  $ DAP$  are erected outside of the convex quadrilateral  $ ABCD$ . Let  $ X$ ,  $ Y$ ,  $ Z$ ,  $ W$  be the midpoints of the segments  $ PQ$ ,  $ QR$ ,  $ RS$ ,  $ SP$ , respectively. Determine the maximum value of
\[ \frac {XZ\plus{}YW}{AC \plus{} BD}.
\]

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