
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 $\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\overrightarrow{AM}=x\overrightarrow{AB}$ and $\overrightarrow{AN}=y\overrightarrow{AC}$ ($xy\neq 0$). Find the minimum value of $4x+y$.

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