
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 $\overrightarrow{a}=(\sqrt{3}\sin x,-\cos x)$, $\overrightarrow{b}=(\cos x,\cos x)$, and the function $f(x)=2\overrightarrow{a}\cdot\overrightarrow{b}+1$.  
$(Ⅰ)$ Find the interval where the function $f(x)$ is monotonically decreasing.  
$(Ⅱ)$ Determine the range of the function $f(x)$ on the interval $[-\frac{\pi}{3},\frac{\pi}{4}]$.

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