
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
Reading material: After learning about inequalities, we can determine the relative size of two numbers or expressions by calculating their difference based on the basic properties of equations and inequalities. If two numbers or expressions are denoted as $m$ and $n$, then:
- When $m \gt n$, we must have $m-n \gt 0$;
- When $m=n$, we must have $m-n=0$;
- When $m \lt n$, we must have $m-n \lt 0.
The reverse is also true, that is:
- When $m-n \gt 0$, we must have $m \gt n$;
- When $m-n=0$, we must have $m=n$;
- When $m-n \lt 0$, we must have $m \lt n.
For example: Compare $a^{2}+1$ with $2a-1$.
Solution: Since $(a^{2}+1)(2a-1)=(a-1)^{2}+1 \gt 0$, we have $a^{2}+1 \gt 2a-1$.
Solving the problems:
1. Fill in the blanks with "$ \gt $" or "$ \lt $": $3-\sqrt{2}$______$4-2\sqrt{2}$;
2. There are two material options for producing a certain product. Plan 1: Use 4 pieces of type $A$ steel plates and 6 pieces of type $B$ steel plates. Plan 2: Use 3 pieces of type $A$ steel plates and 7 pieces of type $B$ steel plates. It is known that the area of one type $A$ steel plate is larger than that of one type $B$ steel plate. If the area of one type $A$ steel plate is $x$ and the area of one type $B$ steel plate is $y$, which plan should be chosen from the perspective of saving materials? Please explain the reason.
3. Given $a \gt 0$, compare $a$ with $\frac{1}{a}$.

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