
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 coordinate system xOy, point A(2, 0) lies on curve C<sub>1</sub> with the parametric equations $$x=acos\varphi,\ y=asin\varphi \quad (a > 0,\ \varphi \text{ is a parameter}).$$ Taking the origin O as the pole and the positive x-axis as the polar axis, the polar equation of curve C<sub>2</sub> is given by $$\rho=acos\theta.$$ 
(Ⅰ) Find the Cartesian equation of curve C<sub>2</sub>.
(Ⅱ) Suppose the polar coordinates of points M and N are $(\rho_{1}, \theta)$ and $(\rho_{2}, \theta+\frac{\pi}{2})$ respectively. If points M and N both lie on curve C<sub>1</sub>, calculate the value of $$\frac{1}{\rho_{1}^{2}} + \frac{1}{\rho_{2}^{2}}.$$

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