
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 $\overrightarrow{a} = (\sqrt{3}\sin 3x, -y)$ and $\overrightarrow{b} = (m, \cos 3x - m)$ (where $m \in \mathbb{R}$), and it is known that $\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{0}$. Let $y = f(x)$.
(1) Find the expression of $f(x)$ and the coordinate of the lowest point M on the graph of $f(x)$ in the interval $\left[\frac{\pi}{18}, \frac{2\pi}{9}\right]$.
(2) If for any $x \in \left[0, \frac{\pi}{9}\right]$, it holds that $f(x) > t - 9x + 1$ for some real number $t$, determine the range of $t$.

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