
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
$(1)$ Solve the inequality $A_6^{x}＜4A_{6}^{x-2}$.  
$(2)$ If $C_{3}^{2}+C_{4}^{2}+C_{5}^{2}+…+C_{n}^{2}=55$, find the positive integer $n$.

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