
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 $0 < x < \frac{\pi}{2}$, simplify the expression: $\lg(\cos x \tan x + 1 - 2 \sin^2{\frac{x}{2}}) + \lg{[\sqrt{2} \cos{(x - \frac{\pi}{4})}]}- \lg{(1 + \sin{2x})}$;
Given $0 < x < 1$ and $x + x^{-1} = 3$, find the value of $x^{\frac{1}{2}} - x^{-\frac{1}{2}}$.

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