
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
On  $xy$ plane the parabola  $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$  intersects with the line  $y=x$  at the point  $P$  that is different from the origin.
Assumed that the circle  $C$  is touched to  $K$  at  $P$  and  $y$  axis at the point  $Q.$ 
Let  $S_{1}$  be the area of the region surrounded by the line passing through two points  $P,\ Q$  and  $K,$  or  $S_{2}$  be the area of the region surrounded by the line which is passing through  $P$  and parallel to  $x$  axis and  $K.$  Find the value of  $\frac{S_{1}}{S_{2}}.$ 

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