
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
A point inside the circle $x^2 + y^2 = 10x$, identified as $(5,3)$, has $k$ chords whose lengths form an arithmetic sequence. The shortest chord length is the first term $a_1$ of the sequence, and the longest chord length is the last term $a_k$. If the common difference $d$ is within the interval $\left[\frac{1}{3}, \frac{1}{2}\right]$, determine the value of $k$.

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