
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 a triangle $\triangle ABC$, where $D$ is the midpoint of side $BC$, and a line passing through point $D$ intersects lines $AB$ and $AC$ at points $E$ and $F$ respectively, and if $\overrightarrow{AE} = \lambda \overrightarrow{AB}$ and $\overrightarrow{AF} = \mu \overrightarrow{AC}$, where $\lambda > 0$ and $\mu > 0$, calculate the minimum value of $\lambda\mu$.

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