
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 that $\Delta ABC$ is a triangle, point $P$ lies on side $BC$ such that $\overrightarrow{BP}=\frac{1}{2}\overrightarrow{PC}$. Points $M$ and $N$ lie on the line passing through $P$ with $\overrightarrow{AM}=\lambda\overrightarrow{AB}$ and $\overrightarrow{AN}=\mu\overrightarrow{AC}$, where $\lambda, \mu > 0$. Determine the minimum value of $\lambda + 2\mu$.

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