
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
Triangle \(ABC\) has sides \(AB = 14\), \(BC = 13\), and \(CA = 15\). It is inscribed in circle \(\Gamma\), which has center \(O\). Let \(M\) be the midpoint of \(AB\), let \(B'\) be the point on \(\Gamma\) diametrically opposite \(B\), and let \(X\) be the intersection of \(AO\) and \(MB'\). Find the length of \(AX\).

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