
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 the parametric equation of curve \( C \) is
\[ \begin{cases} x=2\cos \theta \\ y=3\sin \theta \end{cases} \quad (\theta \text{ is the parameter}), \]
in the same Cartesian coordinate system, by applying the coordinate transformation
\[ \begin{cases} x'= \frac{1}{2}x \\ y'= \frac{1}{3}y \end{cases}, \]
we obtain curve \( C' \).
(1) Find the standard equation of curve \( C' \).
(2) Let point \( A \) be on curve \( C' \) and point \( B \) be at \( (3,0) \). When point \( A \) moves on curve \( C' \), find the equation for the trajectory of the midpoint \( P \) of segment \( AB \).

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