
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
"Jianzhan" is a representative of black porcelain and a business card of Nanping. There are currently two methods of roasting "Jianzhan": "firewood firing" and "electric firing", and the raw materials for making the billet are local red clay and white clay. For a certain type of Jianzhan with the same specifications, the billet of a firewood firing requires 90 grams of red clay and 60 grams of white clay, while the billet of an electric firing requires 75 grams of red clay and 75 grams of white clay. In a production workshop, without considering breakage, the workshop used 1530 grams of red clay and 1170 grams of white clay in a single production.<br/>$(1)$ In this production, how many "firewood firing" and "electric firing" Jianzhan were produced respectively?<br/>$(2)$ The workshop plans to purchase gift boxes. There are two types of gift boxes available: Gift box $A$ can hold 2 Jianzhan, and gift box $B$ can hold 6 Jianzhan. How many purchasing plans are there to exactly pack all the Jianzhan produced this time into gift boxes, and the gift boxes are full? Please explain the reasons.

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