
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 two regular tetrahedra \( A-OBC \) and \( D-OBC \) with coincident bases, points \( M \) and \( N \) are the centroids of triangles \( \triangle ADC \) and \( \triangle BDC \) respectively. Denote \(\overrightarrow{OA} = \boldsymbol{a}, \overrightarrow{OB} = \boldsymbol{b}, \overrightarrow{OC} = \boldsymbol{c}\). If point \( P \) satisfies \(\overrightarrow{OP} = x \boldsymbol{a} + y \boldsymbol{b} + z \boldsymbol{c}\) and \(\overrightarrow{MP} = 2 \overrightarrow{PN}\), find the real numbers \( x \), \( y \), and \( z \).

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