
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 are the points  $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$  in the space. Let  $P_{ij} (i,j \in 0,1,2,3)$  be the point determined by the equality:  $\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}$ . Find the volume of the smallest convex polyhedron which contains all the points  $P_{ij}$ .

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