
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 \(\mu\) be a probability measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}\right)\right)\) with the distribution function \(F = F(x)\), \(x \in \mathbb{R}^{n}\), and the characteristic function \(\varphi = \varphi(t)\), \(t \in \mathbb{R}^{n}\).

Establish the validity of the multidimensional inversion formula

\[ 
\begin{aligned}
\int_{\mathbb{R}^{n}} \prod_{k=1}^{n}\left[I_{\left(a_{k}, b_{k}\right)}+\frac{I_{\left\{a_{k}\right\}}+I_{\left\{b_{k}\right\}}}{2}\right] & d \mu = \\
& = \lim _{c \rightarrow \infty} \frac{1}{(2\pi)^{n}} \int_{[-c, c]^{n}} \prod_{k=1}^{n} \frac{e^{-it_{k}a_{k}}-e^{-it_{k}b_{k}}}{it_{k}} \varphi(t) dt
\end{aligned}
\]

for all \(\left(a_{1}, \ldots, a_{n}\right), \left(b_{1}, \ldots, b_{n}\right) \in \mathbb{R}^{n}\), where \(t = \left(t_{1}, \ldots, t_{n}\right)\).

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