
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, the parametric equations of curve C<sub>1</sub> are:
$$
\begin{cases}
x = a \cos \phi \\
y = b \sin \phi
\end{cases}
$$
where $a > b > 0$ and $\phi$ is the parameter. Using the origin O as the pole and the positive half-axis of the x-axis as the polar axis, establish a polar coordinate system. Curve C<sub>2</sub> is a circle with its center on the polar axis and passing through the pole. It's given that point M(2, $\sqrt{3}$) on curve C<sub>1</sub> corresponds to the parameter $\phi= \frac{\pi}{3}$; and $\theta= \frac{\pi}{4}$; meets curve C<sub>2</sub> at point D($\sqrt{2}$, $\frac{\pi}{4}$).

(1) Find the equations of curves C<sub>1</sub> and C<sub>2</sub>.

(2) Points A($\rho_1$, $\theta$) and B($\rho_2$, $\theta + \frac{\pi}{2}$) lie on curve C<sub>1</sub>. Find 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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