
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
Given two arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ with the sum of their first $n$ terms being $S_{n}$ and $T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=\frac{2n+30}{n+3}$, find the number of values of $n$ that make $\frac{a_{n}}{b_{n}}$ an integer.

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