
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 $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $c \cdot \cos A - a \cdot \cos C = \frac{2}{3}b$.
(1) Find the value of $\frac{\tan A}{\tan C}$;
(2) If $\tan A$, $\tan B$, $\tan C$ form an arithmetic sequence, find the value of $\frac{a^{2} - b^{2} - c^{2}}{bc}$.

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