
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 $q$, let $c(q)$ denote the unique positive integer $m$ such that $|m-\sqrt{q}|<\frac{1}{3}$. For example, $c(6)=2$ and $c(23)=5$. Find $T=\sum_{q=1}^{3000} c(q)$.

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