
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 the graph of the function $y=f(x)$ be obtained by shifting the graph of $y=2\sin 2x+1$ to the left by $\dfrac{\pi}{12}$ units.  
$(1)$ Find the smallest positive period of $f(x)$ and the interval of monotonic increase:  
$(2)$ In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$, respectively, and $f(A)=2$, $b=1$, $S_{\triangle ABC}= \sqrt{3}$. Find the value of $a$.

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