
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 following material and answer the questions. The size of two numbers can be determined by their difference. If two numbers $a$ and $b$ are compared, then:
- When $a \gt b$, it must be $a-b \gt 0$;
- When $a=b$, it must be $a-b=0$;
- When $a \lt b$, it must be $a-b \lt 0$.
The reverse is also true, that is:
- When $a-b \gt 0$, it must be $a \gt b$;
- When $a-b=0$, it must be $a=b$;
- When $a-b \lt 0$, it must be $a \lt b$.
Therefore, comparing the sizes of two quantities, you can first find their difference, and then determine the sizes of the two quantities based on the sign of the difference.
$(1)$ Fill in the blanks with "$ \gt $" or "$ \lt $".
① If $a-b=4$, $a \_\_\_\_\_\_b$;
② If $a-b=-2$, $a \_\_\_\_\_\_b$;
$(2)$ If $x \gt 0$, compare the sizes of $-x+5$ and $-2x+4$;
$(3)$ Compare the sizes of $5x+13y+2$ and $6x+12y+2.

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