
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 $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are non-zero, non-collinear vectors. Let $\overrightarrow{OC} = \frac{1}{1+r} \overrightarrow{OA} + \frac{r}{1+r} \overrightarrow{OB}$. Define the set of points
$$
M = \left\{ K \mid \frac{\mid \overrightarrow{K A} \cdot \overrightarrow{K C}}{|\overrightarrow{K A}|} = \frac{\overrightarrow{K B} \cdot \overrightarrow{K C}}{|\overrightarrow{K B}|} \right\}.
$$

If $K_1, K_2 \in M$ such that for any $r \geq 2$, the inequality $\left|\overrightarrow{K_1 K_2}\right| \leq c|\overrightarrow{AB}|$ always holds, what is the minimum value of the real number $c$?

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