
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 7 and is centered at \(O\). Point \(A\) lies outside \(\omega\) such that \(OA = 15\). The two tangents to \(\omega\) from \(A\) touch it at points \(T_1\) and \(T_2\), leading to points \(B\) and \(C\) on these tangents, respectively, such that line \(BC\) is tangent to \(\omega\) at point \(T_3\) and \(\omega\) lies outside triangle \(ABC\). Given that \(BC = 10\), compute \(AB + AC\).

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