
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
In the right triangle  $ABC$  with shorter side  $AC$  the hypotenuse  $AB$  has  length  $12$ . Denote  $T$  its centroid and  $D$  the feet of altitude from the vertex  $C$ . Determine the size of its inner angle at the vertex  $B$  for which the triangle  $DTC$  has the greatest possible area.

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