
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 a triangle $ABC$ where $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $b$ is the arithmetic mean of $a$ and $c$, and $\operatorname{tg} \frac{B}{2}$ is the geometric mean of $\operatorname{tg} \frac{A}{2}$ and $\operatorname{tg} \frac{C}{2}$, find the quadratic equation whose roots are $\operatorname{tg} \frac{A}{2}$ and $\operatorname{tg} \frac{C}{2}$.

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