
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  $a,\ b$  be positive real numbers. Consider the circle  $C_1: (x-a)^2+y^2=a^2$  and the ellipse  $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which  $C_1$  is inscribed in  $C_2$ .

(2) Suppose  $b=\frac{1}{\sqrt{3}}$  and  $C_1$  is inscribed in  $C_2$ . Find the coordinate  $(p,\ q)$  of the point of tangency in the first quadrant for  $C_1$  and  $C_2$ .

(3) Under the condition in (1), find the area of the part enclosed by  $C_1,\ C_2$  for  $x\geq p$ .

60 point

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