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

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