
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
$(1)$ Calculate: $A_5^1+A_5^2+A_5^3+A_5^4+A_5^5$;<br/>$(2)$ Given $C_5^m=C_5^{2m-1}$, $\left(m \gt 1\right)$; find the value of $C_6^m+C_6^{m+1}+C_7^{m+2}+C_8^{m+3}$.

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