
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 the vector $\overrightarrow{a} = (\sqrt{3}\sin\theta + \cos\theta + 1, 1)$, $\overrightarrow{b} = (1, 1)$, $\theta \in [\frac{\pi}{3}, \frac{2\pi}{3}]$, and $m$ is the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$, find the maximum value of $m$.

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