
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: $3A_{x}^{3}≤2A_{x+1}^{2}+6A_{x}^{2}$;<br/>$(2)$ Find the value of $C_{n}^{5-n}+C_{n+1}^{9-n}$;<br/>$(3)$ Given $\frac{1}{C_{5}^{m}}-\frac{1}{C_{6}^{m}}=\frac{7}{10C_{7}^{m}}$, find $C_{8}^{m}$.

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