
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 from the formulas for $\mathrm{P}\left(B_{m}\right)$ and $\mathrm{P}\left(B_{\geqslant m}\right)$ from problem I.1.13 that for any even $r$ the Bonferroni inequalities hold:

$$
\begin{aligned}
S_{m}+\sum_{k=1}^{r+1}(-1)^{k} C_{m+k}^{k} S_{m+k} \leqslant \mathrm{P}\left(B_{m}\right) & \leqslant S_{m}+\sum_{k=1}^{r}(-1)^{k} C_{m+k}^{k} S_{m+k} \\
S_{m}+\sum_{k=1}^{r+1}(-1)^{k} C_{m+k-1}^{k} S_{m+k} \leqslant \mathrm{P}\left(B_{\geqslant m}\right) & \leqslant S_{m}+\sum_{k=1}^{r}(-1)^{k} C_{m+k-1}^{k} S_{m+k}
\end{aligned}
$$

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