What is the product of the following reaction?

Considering the following reaction sequence,the correct statement$(s)$ is(are):
$A$. Compounds $P$ and $Q$ are carboxylic acids.
$B$. Compound $S$ decolorizes bromine water.
$C$. Compounds $P$ and $S$ react with hydroxylamine to give the corresponding oximes.
$D$. Compound $R$ reacts with dialkylcadmium to give the corresponding tertiary alcohol.

Give the conversion of the following:
$(i)$ $\text{Pent-3-en-2-one}$ from $\text{Ethanal}$
$(ii)$ $\text{Pent-2-enal}$ from $\text{Ethanal}$

$CH_3CHO + HCHO \text{ (excess)} \xrightarrow{\text{Conc. } NaOH} \text{Product}$. The product is:

In the given reaction,
$Cyclohexanone \xrightarrow[(ii) \text{ heat}]{(i) \text{ } OH^-} 'P'$
The number of $\pi$ electrons present in the product $'P'$ is $....$

Propyne was subjected to a reaction with $HgSO_4/dil. H_2SO_4$,which resulted in a product $P$. The product $P$ was heated with $Ba(OH)_2$ to give the product $Q$. The molecular formula of the product $Q$ is