
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 $f(x)=\sin 2x+2 \sqrt{2}(\tan \theta+1)\sin \left(\theta+ \frac{\pi}{4}\right)+\tan^2\theta$, where $x\in\left[- \frac{\pi}{2},0\right],\theta\in\left[- \frac{\pi}{3}, \frac{\pi}{3}\right]$

$(1)$ If $\theta= \frac{\pi}{4}$, find the minimum value of the function $f(x)$ and the corresponding value of $x$;

$(2)$ Determine whether there exists a real number $\theta$ such that the graph of the function $f(x)$ is always above the line $y=-2$. If it exists, find the range of the real number $\theta$; if not, explain why.

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