
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
Circle $\omega$ has a radius of 8 and is centered at $O$. Point $A$ lies outside $\omega$ with $OA=17$. The two tangents to $\omega$ from $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ with $\omega$ lying outside triangle $ABC$. Compute $AB+AC$ given that $BC=12$.

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