
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 $f(x)=\cos^2\left(x+ \frac{\pi}{12}\right)$, $g(x)=1+ \frac{1}{2}\sin 2x$.  
$(I)$ Find the equation of the axis of symmetry for the graph of the function $y=f(x)$.  
$(II)$ Find the smallest positive period and the range of the function $h(x)=f(x)+g(x)$.

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