
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 $\overrightarrow{a} = (\sqrt{3}, \cos 2\omega x)$ and $\overrightarrow{b} = (\sin 2\omega x, 1)$ with $\omega > 0$, let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$ and the period of $f(x)$ is $\pi$.
1. Find the analytical expression for function $f(x)$.
2. If $x \in [0, \frac{\pi}{2}]$ and $f(x) + m \leq 3$, find the range of real number $m$.

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