
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
Cube \(ABCD EFGH\) has an edge length of 100. Point \(P\) is on \(AB\), point \(Q\) is on \(AD\), and point \(R\) is on \(AF\) such that \(AP = x\), \(AQ = x + 1\), and \(AR = \frac{x + 1}{2x}\) for some integer \(x\). For how many integers \(x\) is the volume of the triangular-based pyramid \(APQR\) between 0.04% and 0.08% of the volume of cube \(ABCDEFGH\)? (The volume of a pyramid is equal to one-third of the area of its base times its height.)

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