
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. If $\cos 2B + \cos B + \cos (A-C) = 1$ and $b = \sqrt{7}$, find the minimum value of $a^2 + c^2$.

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