
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 curve \( C_1: y = \sqrt{-x^2 + 10x - 9} \) and the point \( A(1,0) \). If there exist distinct points \( B \) and \( C \) on the curve \( C_1 \) such that their distances to the line \( l: 3x + 1 = 0 \) are equal to \( |AB| \) and \( |AC| \) respectively, then \( |AB| + |AC| = \quad \) .

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