
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
Using the numbers $1$, $2$, $3$, $4$, $5$ to form a five-digit number $\overline{abcde}$ without repeating digits, where $\overline{abcde}$ represents the digits in the order of ten thousands, thousands, hundreds, tens, and units, determine the number of five-digit numbers that satisfy $a > b > c < d < e$. Find the coefficient of $x^{2}$ in the expansion of $1+\left(1+x\right)^{1}+\left(1+x\right)^{2}+\left(1+x\right)^{3}+\cdots +\left(1+x\right)^{n}$.

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