
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 plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c} (\overrightarrow{c} \neq \overrightarrow{0})$ satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, and $(\overrightarrow{a} - \overrightarrow{b}) \cdot \overrightarrow{c} = 0$. Let the projection of plane vector $\overrightarrow{d}$ in the directions of $\overrightarrow{a}$ and $\overrightarrow{b}$ be $x$ and $y$, respectively, and the projection of $\overrightarrow{d} - \overrightarrow{a}$ in the direction of $\overrightarrow{c}$ be $z$. Find the minimum value of $x^{2} + y^{2} + z^{2}$.

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