
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
Find  $k$  if  $P,Q,R,$  and  $S$  are points on the sides of quadrilateral  $ABCD$  so that \[ \frac{AP}{PB} = \frac{BQ}{QC} = \frac{CR}{RD} = \frac{DS}{SA} = k,  \] and the area of the quadrilateral  $PQRS$  is exactly 52% of the area of the quadrilateral  $ABCD$ .

For picture, go [here](http://www.cms.math.ca/Competitions/IMTS/imts3.html).

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