
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  $C$  be a fixed unit circle in the cartesian plane. For any convex polygon  $P$  , each of whose sides is tangent to  $C$ , let  $N( P, h, k)$  be the number of points common to  $P$  and the unit circle with center at  $(h, k).$  Let  $H(P)$  be the region of all points  $(x, y)$  for which  $N(P, x, y) \geq 1$  and  $F(P)$  be the area of  $H(P).$  Find the smallest number  $u$  with  $$  \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u $$  for all polygons  $P$ , where the double integral is taken over  $H(P).$ 

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