
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
In $\triangle ABC$, $B(-\sqrt{5}, 0)$, $C(\sqrt{5}, 0)$, and the sum of the lengths of the medians on sides $AB$ and $AC$ is $9$.
(Ⅰ) Find the equation of the trajectory of the centroid $G$ of $\triangle ABC$.
(Ⅱ) Let $P$ be any point on the trajectory found in (Ⅰ), find the minimum value of $\cos\angle BPC$.

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