
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
Example 5. Given the probability density function of a two-dimensional random variable $(X, Y)$
$$
\begin{aligned}
& p(x, y)=0.5 \sin (x+y) \quad\left(0 \leq x \leq \frac{\pi}{2}, 0 \leq y \leq \frac{\pi}{2}\right), \\
& p(x, y)=0 \quad \text{outside the square} \quad S=\left[0, \frac{\pi}{2} ; 0, \frac{\pi}{2}\right].
\end{aligned}
$$

Find the cumulative distribution function $F(x, y)$ of this variable. Calculate the probability that $X$ and $Y$ take the values: $X<\frac{\pi}{6}, Y<\frac{\pi}{6}$.

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