
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 point $D$ lies in the plane determined by triangle $ABC$, and $O$ is an arbitrary point outside plane $ABC$. Real numbers $x$ and $y$ satisfy $\overrightarrow{OD}=3\overrightarrow{OC}-x\overrightarrow{OA}-2y\overrightarrow{OB}$. Find the minimum value of $x^{2}+2y^{2}$.

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