
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
All Grade 10 students performed exceptionally well during the fall semester's New Campus Experience Week. The school plans to honor Grade 10;

(1) How many different ways are there to select 3 outstanding classes if one class must be chosen from 6 liberal arts classes, and two classes must be chosen from 14 science classes?

(2) The grade group plans to select 5 students from the chosen three classes, with at least 1 and at most 2 students from each class. How many different allocation plans are there?

(3) The selected 5 students and three grade supervisors, Principal Xu, Principal Chen, and Principal Fu, will line up for a group photo. The teachers cannot stand at either end, and their order is fixed. How many different arrangements are there?

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