
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
For each positive integer $p$, let $a(p)$ denote the unique positive integer $j$ such that $|j-\sqrt[3]{p}|<\frac{1}{2}$. For example, $a(7)=2$ and $a(64)=4$. Find $T=\sum_{p=1}^{500} a(p)$.

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