
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 that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, and $\overrightarrow{a}⋅\overrightarrow{b}=3$, vector $\overrightarrow{c}$ satisfies $\overrightarrow{c}=λ\overrightarrow{a}+({1-λ})\overrightarrow{b}$ ($0＜λ＜1$), and $\overrightarrow{a}⋅\overrightarrow{c}=\overrightarrow{b}⋅\overrightarrow{c}$. Let $x=\frac{{\overrightarrow{c}⋅\overrightarrow{a}}}{{|{\overrightarrow{a}}|}}$ and $y=\frac{{\overrightarrow{c}⋅\overrightarrow{b}}}{{|{\overrightarrow{b}}|}}$. Find the maximum value of $x^{2}+y^{2}-xy$.

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