
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
For the quadratic equation $ax^2+bx+c=0$ ($a \neq 0$) with two real roots $x_1$ and $x_2$, there is a theorem stating that $x_1 + x_2 = -\frac{b}{a}$ and $x_1 \cdot x_2 = \frac{c}{a}$. This theorem is known as Viète's formulas. Using these formulas, solve the following problem: Given $\lg m$ and $\lg n$ are the two real roots of the equation $2x^2 - 4x + 1 = 0$, then  
(1) Find the value of $mn$;  
(2) Find the value of $\log_n m + \log_m n$.

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