
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 triangle  $ABC,$   $\sin \angle A=\frac{4}{5}$  and  $\angle A<90^\circ$   Let  $D$  be a point outside triangle  $ABC$  such that  $\angle BAD=\angle DAC$  and  $\angle BDC = 90^{\circ}.$  Suppose that  $AD=1$  and that  $\frac{BD} {CD} = \frac{3}{2}.$  If  $AB+AC$  can be expressed in the form  $\frac{a\sqrt{b}}{c}$  where  $a,b,c$  are pairwise relatively prime integers, find  $a+b+c$ .

*Author: Ray Li*

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