
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
Read the materials first before solving the problem.

Material One: The method of completing the square can be used to solve quadratic equations in one variable. For example, the equation $x^2 + 2x - 1 = 0$ can be solved by first completing the square to get $(x+1)^2 = 2$, and then applying the square root method to find the solution. Indeed, completing the square can be used to solve many problems.

Material Two: For the algebraic expression $3a^2 + 1$, since $3a^2 \geq 0$, it follows that $3a^2 + 1 \geq 1$, which means $3a^2 + 1$ has a minimum value of 1, and it occurs when $a=0$.

Similarly, for the algebraic expression $-3a^2 + 1$, since $-3a^2 \leq 0$, it follows that $-3a^2 + 1 \leq 1$, which means $-3a^2 + 1$ has a maximum value of 1, and it occurs when $a=0$.

Answer the following questions:
(1) Fill in the blanks: ① When $x=$____, the algebraic expression $2x^2 - 1$ has a minimum value of ____; 
② When $x=$____, the algebraic expression $-2(x+1)^2 + 1$ has a maximum value of ____.
(2) Determine the minimum value of the algebraic expression $2x^2 - 4x + 1$ and the value of $x$ when the expression takes its minimum value. 
(You are required to show the necessary computational reasoning process.)

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