
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  $ABC$  with  $AC=20$  is inscribed in a circle  $\omega$ . A tangent  $t$  to  $\omega$  is drawn through  $B$ . The distance  $t$  from  $A$  is  $25$  and that from  $C$  is  $16$ .If  $S$  denotes the area of the triangle  $ABC$ , find the largest integer not exceeding  $\frac{S}{20}$ 

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