
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
Find the range of the following functions:
(1) $y=x+\sqrt{1-2x}$;
(2) $y=\frac{10^x + 10^{-x}}{10^x - 10^{-x}}$;
(3) $y=\log_{0.5}(2x - x^2 + 3)$;
(4) $y=\sqrt{x^2 + 2x + 2} + \sqrt{x^2 - 2x + 2}$;
(5) $y=|x+1| + |x-1| + |x+2|$;
(6) $y=\frac{x^2 + 4x + 3}{x^2 + x - 6}$;
(7) $y=x^2 + \frac{1}{x}$ where $x \in \mathbb{R}^+$.

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