
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
In the Cartesian plane, let  $S_{i,j} = \{(x,y)\mid i \le x \le j\}$ . For  $i=0,1,\ldots,2012$ , color  $S_{i,i+1}$  pink if  $i$  is even and gray if  $i$  is odd. For a convex polygon  $P$  in the plane, let  $d(P)$  denote its pink density, i.e. the fraction of its total area that is pink. Call a polygon  $P$  *pinxtreme* if it lies completely in the region  $S_{0,2013}$  and has at least one vertex on each of the lines  $x=0$  and  $x=2013$ . Given that the minimum value of  $d(P)$  over all non-degenerate convex pinxtreme polygons  $P$  in the plane can be expressed in the form  $\frac{(1+\sqrt{p})^2}{q^2}$  for positive integers  $p,q$ , find  $p+q$ .

*Victor Wang.*

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