
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. The coefficient of ${x^4}$ in the expansion of the binomial ${(x-\frac{2}{x})^6}$ is \_\_\_\_\_\_\_\_\_\_.
2. Let the sum of the first $n$ terms of the sequence ${{a_n}}$ be ${S_n}$, with ${{a_1}=1}$ and ${{a_{n+1}}=2{S_n}+3}$. Find ${S_5}$ equals \_\_\_\_\_\_\_\_\_\_.
3. Given an equilateral triangle $ABC$ with side length $3$ and all three vertices on the sphere with center $O$, the volume of the triangular pyramid $O-ABC$ is $\frac{3}{2}\sqrt{3}$. Find the surface area of the sphere \_\_\_\_\_\_\_\_\_\_.
4. Given functions $f(x)=\lg ({x^2}+1)$ and $g(x)=( \frac{1}{2}{)}^{x}-m$, for any ${{x_1}∈[0,3]}$, there exists ${{x_2}∈[1,2]}$ such that $f({{x_1}})\leqslant g({{x_2}})$. Find the range of the real number $m$ \_\_\_\_\_\_\_\_\_\_.

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