
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
1. Evaluate the expression:
$$\lg 5 + \lg 2 + \left(\frac{3}{5}\right)^0 + \ln e^{\frac{1}{2}}.$$
*(where $e$ is the base of the natural logarithm)*;

2. Given $\cos \alpha = \frac{2\sqrt{2}}{3}$ and $\sin(\alpha + \beta) = \frac{1}{3}$, with $\alpha \in \left(0, \frac{\pi}{2}\right)$ and $\beta \in \left(\frac{\pi}{2}, \pi\right)$, find the value of $\cos \beta$.

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