
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
$P$ is a point inside or on the surface of a unit cube, satisfying the following conditions:
(1) One edge of the cube has two endpoints with distances from $P$ being $\frac{8}{15}$ and $\frac{17}{15}$ respectively.
(2) At least two vertices of the cube have equal distances to $P$.
Determine the number of points $P$ that satisfy the above conditions.

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