
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
Inside the circle $x^{2}+y^{2}-5 x=0$, passing through the point $\left(\frac{5}{2}, \frac{3}{2}\right)$, there are $k (\geqslant 3)$ chords whose lengths form an arithmetic sequence. If the common difference $d$ belongs to $\left(\frac{1}{6}, \frac{1}{3}\right]$, determine the possible values of $k$.

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