
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 vectors $\overrightarrow{a}=( \sqrt{3}\sin x,\cos x+\sin x)$, $\overrightarrow{b}=(2\cos x,\sin x-\cos x)$, and a function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$:
1. Find the intervals where the function $f(x)$ is monotonically increasing or decreasing.
2. When $x \in [\frac{5\pi}{24}, \frac{5\pi}{12}]$, find the range of real values for $m$ such that the inequality $mt^{2}+mt+3\geqslant f(x)$ holds for any $t \in \mathbb{R}$.

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