
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
Call a positive integer $n$ $m$-pretty if $n$ has exactly $m$ positive divisors, and $n$ is divisible by $m$. Specifically, consider $n = 2^a 7^b k$, where $a, b \geq 1$, $k$ is coprime with $14$, and $d(n) = 18$ (the divisor function). Find the sum $T$ of all positive integers less than $1500$ that are $18$-pretty, and determine $\frac{T}{18}$.

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