
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 an equilateral triangle $ABC$ with side length $1$, $\overrightarrow{BD}=x \overrightarrow{BA}$, $\overrightarrow{CE}=y \overrightarrow{CA}$, where $x > 0$, $y > 0$, and $x+y=1$, find the maximum value of $\overrightarrow{CD}\cdot \overrightarrow{BE}$.

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