
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
In the  $ xyz$  space with the origin  $ O$ , given a cuboid  $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$  and the plane  $ \alpha : z \equal{} 2$ . Draw the perpendicular  $ PH$  from  $ P$  to the plane. Find the volume of the solid formed by all points of  $ P$  which are included in  $ K$  such that  $ \overline{OP}\leq \overline{PH}$ .

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