
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. If $(1+2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}$, then $a_{0}+a_{2}+a_{4}=$ \_\_\_\_\_\_.
2. A school has selected five teachers: A, B, C, D, and E to teach at three distant schools. Each school must have at least one teacher. Teachers A and B must teach at the same school, while teachers A and C must teach at different schools. How many different arrangements are possible?
3. The constant term of the expansion of $(x-\frac{1}{x})(2x+\frac{1}{x})^{5}$ is \_\_\_\_\_\_.
4. If the line $y=kx+b$ is a tangent line to both the curve $y=\ln x+2$ and the curve $y=\ln(x+1)$, then $b=$ \_\_\_\_\_\_.

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