
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 $F$ is any point (excluding endpoints) on segment $BC$. If $\overrightarrow{AF}=x\overrightarrow{AB}+2y\overrightarrow{AC}\left(x \gt 0,y \gt 0\right)$, then the minimum value of $\frac{1}{x}+\frac{2}{y}$ is ______.

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