
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 is a non-isosceles triangle  $ABC$  with  $\angle ABC=60^{\circ}$ , and in its interior, a point  $T$  is selected such that  $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$ . Let  $M$   the intersection point of the medians in  $ABC$ . Let  $TM$  intersect  $(ATC)$  at  $K$ . Find  $TM/MK$ .

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