
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
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$. In the interval $[-1,1]$, $f(x)$ is defined as follows: $$f(x) = \begin{cases} ax+1 & ;-1\leqslant x < 0 \\ \frac{bx+2}{x+1} & ;0\leqslant x\leqslant 1 \end{cases}$$ where $a$, $b\in\mathbb{R}$. If $f(\frac{1}{2})=f(\frac{3}{2})$, find the value of $a-2b$.

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