
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 that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtuse, the minimum value of $|\overrightarrow{b} - t\overrightarrow{a}| (t \in \mathbb{R})$ is $\frac{\sqrt{3}}{2}$, and $(\overrightarrow{c} - \overrightarrow{a}) \cdot (\overrightarrow{c} - \overrightarrow{b}) = 0$, find the maximum value of $\overrightarrow{c} \cdot (\overrightarrow{a} + \overrightarrow{b})$.

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