
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 that within the circle $x^2+y^2=5x$, there are $n$ chords passing through the point $\left( \frac {5}{2}, \frac {3}{2}\right)$ whose lengths form an arithmetic sequence, with the shortest chord length as the first term $a_1$ of the sequence and the longest chord length as $a_n$, and the common difference $d$ is in the interval $\left( \frac {1}{6}, \frac {1}{3}\right]$, determine the set of possible values for $n$.

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