
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
A line passing through point A (0, $a$) intersects circle M: $(x-2)^2+y^2=1$ at points B and C.  
(Science) Take a point P on line segment BC such that $\overrightarrow {AB}=\lambda \overrightarrow {AC}$ and $\overrightarrow {BP}=\lambda \overrightarrow {PC}$, where $\lambda \in \mathbb{R}$.  
(Arts) Take a point P on line segment BC such that the reciprocals of the x-coordinates of points B, P, and C form an arithmetic sequence.  
(1) Find the equation of the trajectory of point P.  
(2) If the trajectory from (1) intersects circle M at points R and S, find the maximum value of the area of $\triangle MRS$.

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