
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 $\overrightarrow{a}=(\sqrt{3}\sin x,\cos x+\sin x)$, $\overrightarrow{b}=(2\cos x,\sin x-\cos x)$, $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.  
$(1)$ Find the monotonic intervals of the function $f(x)$.  
$(2)$ When $x\in\left[\frac{5\pi}{24}, \frac{5\pi}{12}\right]$, for any $t\in\mathbb{R}$, the inequality $mt^{2}+mt+3\geqslant f(x)$ always holds, find the range of the real number $m$.

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