
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  $ABCD$  be a convex quadrilateral. Construct equilateral triangles  $AQB$ ,  $BRC$ ,  $CSD$  and  $DPA$  externally on the sides  $AB$ ,  $BC$ ,  $CD$  and  $DA$  respectively. Let  $K, L, M, N$  be the mid-points of  $P Q, QR, RS, SP$ . Find the maximum value of  $$ \frac{KM + LN}{AC + BD} $$  .

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