
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
Place point \( C \) at the origin of the Cartesian coordinate system and point \( B \) on the x-axis, as shown in the diagram. Then the coordinates of points \( C, B, A \) will be \( C(0,0), B(3,0), A(-\frac{1}{2}, \frac{\sqrt{3}}{2}) \). The locus of points \( K(x, y) \) such that \( \frac{BK}{CK} = 1:3 \) is a circle. Indeed,

\[
\frac{BK^2}{CK^2} = \frac{(x-3)^2 + y^2}{x^2 + y^2} = \frac{1}{9}
\]

which is equivalent to

\[
\left(x - \frac{27}{8}\right)^2 + y^2 = \frac{81}{64}
\]

Similarly, the locus of points \( M \) such that \( \frac{AM}{MC} = 2:3 \) is a circle given by the equation

\[
\left(x + \frac{9}{10}\right)^2 + \left(y - \frac{9\sqrt{3}}{10}\right)^2 = \frac{36}{25}
\]

To maximize the distance between points \( K \) (lying on circle (1)) and \( M \) (on circle (2)), these points should be placed on the line passing through the centers of the circles. Thus, the maximum distance sought is equal to the distance between the centers of circles (1) and (2) plus the lengths of their radii:

\[
MK_{\max} = \frac{9}{8} + \frac{6}{5} + \frac{9\sqrt{409}}{40} = \frac{93 + 9\sqrt{409}}{40}
\]

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