
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
Given a polar coordinate system where the pole is at the origin of the rectangular coordinate system, the polar axis is the positive half of the x-axis, and the unit lengths are the same. There are two curves $C_1: \rho = 1$ and $C_2$ with the parametric equations $\begin{cases} x = \frac{\sqrt{2}}{2}t - 1 \\ y = \frac{\sqrt{2}}{2}t + 1 \end{cases}$ where $t$ is the parameter.

(1) Find the minimum distance from a point on curve $C_1$ to curve $C_2$.
(2) If we expand the horizontal coordinates of all points on curve $C_1$ to twice their original value and the vertical coordinates to $\sqrt{3}$ times their original value, we obtain curve $C_3$. Let $P(-1, 1)$, and suppose curves $C_2$ and $C_3$ intersect at points $A$ and $B$. Find $|PA| + |PB|$.

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