
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
Derive the formula

$$
I_{i}^{*}=\frac{b-a}{n} \sum_{i=1}^{n}\left[\varphi\left(x_{i}\right)-g\left(x_{i}\right)\right]+\int_{a}^{b} g(x) \mathrm{d} x
$$

where $x_{i}=a+(b-a)r_{i}$, and $g(x) \simeq \varphi(x)$, for the estimation of the integral

$$
I=\int_{a}^{b} \varphi(x) \mathrm{d} x
$$

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