
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 an ellipse $C$ with its center at the origin, its left focus at $F(-\sqrt{3}, 0)$, and its right vertex at $A(2, 0)$. Let point $B(3, 0)$.
(1) Find the standard equation of the ellipse $C$;
(2) If $P$ is a moving point on the ellipse $C$, find the equation of the trajectory of the midpoint $M$ of the line segment $PB$.

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