
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 function $f(x) = 2\cos^2x + 2\sqrt{3}\sin x\cos x + m$.

(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic decrease;  
(2) If $x \in \left[0, \frac{\pi}{2}\right]$, does there exist a real number $m$ such that the range of the function $f(x)$ is exactly $\left[\frac{1}{2}, \frac{7}{2}\right]$? If it exists, find the value of $m$; if not, explain why.

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