
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 a circle $(x)^{2}+(y)^{2}-6x-8y+21=0$ and a line $kx-y-4k+3=0$.

$(1)$ If the line always intersects the circle at two distinct points, find the set of values for $k$.

$(2)$ Find the value of $k$ for which the chord cut by the line from the circle is the shortest, and calculate the length of this shortest chord.

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