
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 point $A(0,a)$, a line intersects circle $M$: $(x-2)^{2}+y^{2}=1$ at points $B$ and $C$. Take a point $P$ on $BC$ such that $\overrightarrow{AB}=λ \overrightarrow{AC}$ and $\overrightarrow{BP}=λ \overrightarrow{PC},(λ∈R)$. Additionally, 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 trajectory equation of point $P$.
2. If the trajectory intersects circle $M$ at points $R$ and $S$, find the maximum area of $△MRS$.

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