
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. On the interval $[-1, 1]$, $f(x)$ is defined as:

$$
f(x) =
\begin{cases}
ax+1 & \text{for } -1 \leq x < 0, \\
\frac{bx+2}{x+1} & \text{for } 0 \leq x \leq 1,
\end{cases}
$$

where $a, b \in \mathbb{R}$. If $f\left(\frac{1}{2}\right) = f\left(\frac{3}{2}\right)$, then $3a+2b =$ ______.

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