
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 the condition that there exist exactly  $1990$  triangles  $ABC$  with integral side-lengths satisfying the following conditions:
(i)  $\angle ABC =\frac 12 \angle BAC;$ (ii)  $AC = b.$ Find the minimal value of  $b.$ 

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