
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
[IMO 2007 HKTST 1](http://www.mathlinks.ro/Forum/viewtopic.php?t=107262)
Problem 2

Let  $A$ ,  $B$  and  $C$  be real numbers such that
(i)  $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$ ,
(ii)  $\tan C$  and  $\cot C$  are defined.
Find the minimum value of  $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$ .

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