
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 that the three interior angles \( A \), \( B \), and \( C \) of triangle \(\triangle ABC\) satisfy the equations \( A + C = 2B \) and \(\frac{1}{\cos A} + \frac{1}{\cos C} = -\frac{\sqrt{2}}{\cos B} \), find \(\cos \frac{A - C}{2}\).

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