
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
Within the circle $x^2+y^2=5x$, there are $n$ chords passing through the point $(\frac{5}{2},\frac{3}{2})$ with lengths forming an arithmetic sequence. The shortest chord length is the first term of the sequence, $a_1$, and the longest chord length is the last term, $a_n$. If the common difference $d\in[\frac{1}{6},\frac{1}{3}]$, what are the possible values for $n$?

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