Why do these reactions occur? Identify the products $[A]$ to $[D]$:
  
  $\ce{CH3COONa}+\ce{NaNH2} + \triangle \rightarrow [A]$
  
  $[A] +\ce{CO_2} \rightarrow [B]$
  
  $[B] + \ce{H2O} \rightarrow [C]$
  
  $[C] +\ce{P2O5 } \rightarrow [D]$
  
  $[A],[B],[C],[D]$ are the main product in each step.


According to my book's answer 

$[A]=\ce{NaCH2COONa}$

$[B]=\ce{NaOOC-COONa}$

$[C]= \ce{(CH2COOH)}_2$

$[D]= \ce{C_3O_2}$

I can intuitively understand that product $[A]$ is correct.However I never learnt the next three reactions.Can someone explain/give a source from which I can learn why the next three reactions occur ?

Thanks.
Abstraction of $\ce{H+}$ by strong base $\ce{NH2^-}$
$$\ce{CH3-COO- + NH2^- -&gt; ^-CH2-COO- + NH3}$$

Carboxylation to form malonate
$$\ce{^-CH2-COO- + CO2 -&gt; ^-OOC-CH2-COO-}$$

Free malonic acid
$$\ce{^-OOC-CH2-COO- + 2 H+ -&gt; HOOC-CH2-COOH}$$

Dehydration of malonic acid with $\ce{P4O10}$ yields carbon suboxide (unbalanced)
$$\ce{HOOC-CH2-COOH + P4O10-&gt; O=C=C=C=O + H3PO4}$$