
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 $\triangle ABC$, points $E$ and $F$ are on $AB$ and $AC$ respectively, with $AE:EB=AF:FC=1:2$. Let $P$ be any point on $EF$, and let real numbers $x$ and $y$ satisfy $\overrightarrow{PA} + x\overrightarrow{PB} + y\overrightarrow{PC} = \overrightarrow{0}$. Let the areas of $\triangle ABC$, $\triangle PBC$, $\triangle PCA$, and $\triangle PAB$ be $S$, $S_1$, $S_2$, and $S_3$ respectively. Define $\frac{S_1}{S} = \lambda_1$, $\frac{S_2}{S} = \lambda_2$, and $\frac{S_3}{S} = \lambda_3$. Find the value of $2x+y$ when the product $\lambda_2 \cdot \lambda_3$ reaches its maximum value.

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