
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 $c(p)$ denote the unique positive integer $k$ such that $|k-\sqrt[3]{p}|<\frac{1}{2}$. For example, $c(7)=2$ and $c(63)=4$. Calculate $T=\sum_{p=1}^{1729} c(p)$.

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