
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 $f(x)=\sqrt{3}\sin(2x+φ)+\cos(2x+φ)$, where $|φ|＜\frac{π}{2}$, the graph of $f(x)$ is shifted to the left by $\frac{π}{3}$ units, resulting in a function whose graph is symmetric about the $y$-axis.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ If the equation $f(x)=a$ has exactly two real roots in the interval $[\frac{π}{6},\frac{5}{12}π]$, find the range of real values for $a$.

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