
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 triangle  $ABC$  is inscribed in a circle  $G$ . For any point  $M$  inside the triangle,  $A_1$  denotes the intersection of the ray  $AM$  with  $G$ . Find the locus of point  $M$  for which  $\frac{BM\cdot CM}{MA_1}$  is minimal, and find this minimum value.

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