
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 $[-1,1]$, $f(x)=\begin{cases} ax+1, & -1\leqslant x < 0, \\ \frac{bx+2}{x+1}, & 0\leqslant x\leqslant 1, \end{cases}$ where $a$, $b\in\mathbb{R}$. If $f\left( \frac{1}{2} \right)=f\left( \frac{3}{2} \right)$, then the value of $a+3b$ is _______.

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