
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 $f(x)=a\cos \omega x$ (a≠0, ω>0), if the graph of the function y=f(x) is shifted to the left by $\frac{\pi}{6\omega}$ units to obtain the graph of the function y=g(x), and if the equation g(x)=0 has exactly two distinct real roots in the interval [0,$\frac{7\pi}{12}$], determine the range of the real number ω.

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