
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 \( \triangle ABC \), \( BC = 4 \) and \( \angle BAC = 60^\circ \). Let \( I \) be the incentre of \( \triangle ABC \). The circle passing through \( B, I \), and \( C \) meets the perpendicular bisector of \( BC \) at a point \( X \) inside \( \triangle ABC \). Find \( AX \).

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