
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
Given the function $f(x)=3\sin (2x+\varphi)$ where $\varphi \in \left(0, \frac{\pi}{2} \right)$, its graph is translated left by $\frac{\pi}{6}$ units and is symmetric about the $y$-axis.

$(1)$ Find the expression for the function $f(x)$.

$(2)$ Explain how its graph is obtained from the graph of $y=\sin x$ through transformations.

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