
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 $\triangle ABC$, point $M$ lies on the line containing side $BC$, and $\overrightarrow{BM}=2\overrightarrow{BC}$. Point $P$ lies on line $AM$. If $\overrightarrow{BP}=\lambda \overrightarrow{BA}+\mu \overrightarrow{BC}$ ($\lambda\ \ \gt 0$, $\mu\ \ \gt 0$), find the minimum value of $\frac{1}{\lambda}+\frac{2}{\mu}$.

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