
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
Let $y=f(x)$ be a continuous function on the interval $[0,1]$ and always satisfies $0 \leq f(x) \leq 1$. The integral $$\int_{0}^{1}f(x)dx$$ can be approximately calculated using the random simulation method. First, generate two sets (each containing $N$ numbers) of uniform random numbers $x_1, x_2, \ldots, x_N$ and $y_1, y_2, \ldots, y_N$ on the interval $[0,1]$, thus obtaining $N$ points $(x_i, y_i)$ ($i=1,2,\ldots,N$). Then, count the number of points $N_1$ among them that satisfy $y_i \leq f(x_i)$ ($i=1,2,\ldots,N$). Therefore, the approximate value of the integral $$\int_{0}^{1}f(x)dx$$ obtained by the random simulation scheme is \_\_\_\_\_\_.

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