
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 $xOy$, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x = -1 + \sqrt{2}t, \\
y = 1 + \sqrt{2}t,
\end{cases}
$$
where $t$ is the parameter. Curve $C_2$ is a circle with center at $(1,2)$ and radius $2$. With the origin as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established.

(1) Find the Cartesian equation of $C_1$ and the polar equation of $C_2$;

(2) Suppose curve $C_3$ passes through the origin and is parallel to curve $C_1$, and it intersects $C_2$ at points $A$ and $B$. Find the value of $\frac{1}{|OA|} + \frac{1}{|OB|}$.

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