
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
The $O x$ and $O y$ axes are perpendicular to each other. Point $C$ lies on $O y$, and points $A$ and $B$ lie on $O x$. Determine the position of point $M$ on $O x$ such that $\frac{M A \cdot M B}{\overline{M C}^{2}}$ attains an extremal value. (Special case: $\overline{O C}^{2} = O A \cdot O B$).

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