
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:
$$
f(x) =
\begin{cases}
ax+1, & \text{if $-1 \leq x < 0$} \\
\frac{bx+2}{x+1}, & \text{if $0 \leq x \leq 1$}
\end{cases}
$$
where $a, b \in \mathbb{R}$. If $f(\frac{1}{2}) = f(\frac{3}{2})$, find the value of $a + 3b$.

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