Perverse sheaves on the complex affine line

Show that a perverse sheaf on $$\mathbb{A}^1(\mathbb{C})$$ (the complex plane with the analytic topology) is a bounded complex $$A$$ of sheaves of $$\mathbb{Q}$$-vector spaces with constructible cohomology sheaves $$\mathcal{H}^n(A)$$ such that the following three conditions hold true:

1. $$\mathcal{H}^n(A)=0$$ unless $$n\in \{-1,0\}$$,
2. $$\mathcal{H}^{-1}(A)$$ has no nonzero global sections with finite support,
3. $$\mathcal{H}^0(A)$$ is a finite sum of skyscraper sheaves.

Although it seems to be a well-known fact, I have some trouble proving it.

My attempt so far. By definition of a perverse sheaf, the following two conditions hold true: for all integers $$q$$, we have:

$$(i)$$ $$\dim \operatorname{supp} \mathcal{H}^{-q}(A)\leq q$$,

$$(ii)$$ $$\dim \operatorname{supp} \mathcal{H}^{-q}(\mathcal{D}A)\leq q$$,

where $$\mathcal{D}A$$ is the Verdier dual of $$A$$ (we agree that the dimension of the empty set is $$-\infty$$). From $$(i)$$, $$q=0$$, we deduce point 3. For $$q<0$$, we deduce half of point 1, that is $$\mathcal{H}^{n}(A)=0$$ for $$n>0$$. Because $$\mathbb{A}^1_{\mathbb{C}}$$ is smooth of dimension $$1$$, we have $$\mathcal{D}A=R\mathcal{Hom}(A,\mathbb{Q}_{\mathbb{A}^1})[2]$$ where $$\mathbb{Q}_{\mathbb{A}^1}$$ is the constant sheaf equal to $$\mathbb{Q}$$ on $$\mathbb{A}^1$$ placed in degree $$0$$. Hence $$\mathcal{H}^{-q}(\mathcal{D}A)=\mathcal{H}^{2-q}(R\mathcal{Hom}(A,\mathbb{Q}_{\mathbb{A}^1})).$$ However, $$\mathbb{Q}_{\mathbb{A}^1}$$ is not an injective complex and nothing ensures me that $$\mathcal{H}^{2-q}$$ commutes with $$R\mathcal{Hom}(-,\mathbb{Q}_{\mathbb{A}^1})$$. How should I pursue my computation?

Many thanks!

It sounds like you are stuck computing the stalks of $$\mathcal D A$$. To do this, you can use that the homology of the stalk of $$\mathcal D A$$ at $$p$$ is the dual of $$H^*(X, X-p ; A) = H^*(U,U-p;A)$$ for any neighborhood $$U$$ of $$p$$. This follows from $$i^*\mathcal D A = \mathcal D i^! A$$ and the exact triangle $$i_* i^! A \to A \to j_* j^* A,$$ where $$i$$ is the inclusion of $$p$$ and $$j$$ is the inclusion of the complement.