
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
Let  $P$  be a square pyramid whose base consists of the four vertices  $(0, 0, 0), (3, 0, 0), (3, 3, 0)$ , and  $(0, 3, 0)$ , and whose apex is the point  $(1, 1, 3)$ . Let  $Q$  be a square pyramid whose base is the same as the base of  $P$ , and whose apex is the point  $(2, 2, 3)$ . Find the volume of the intersection of the interiors of  $P$  and  $Q$ .

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