Calculating the Fundamental Group of a CW Complex with Attaching Maps of Varying Degrees

I’m reviewing old homework questions for an upcoming topology exam, and I came across a question that I did not fully understand at the time. In the solution to this homework question, I had to compute the fundamental group of the following $CW$ complex.
$X$ is the CW complex obtained from $S^1$ with its usual cell structure by attaching two 2-cells by maps of degrees 2 and 3 respectively.

I approached the problem as follows.
Let $U$ be an small open neighborhood of $S^1subset X$ unioned with the first 2 cell attached, and let $V$ be the same small open neighborhood of $S^1subset X$ unioned with the second 2-cell
attached. Then $Ucap V$ is that small open neighborhood of $S^1$, so we get that $U,V,Ucap V$ and are open and path connected, and $X=Ucup V$. I wish to apply SVK’s theorem to $U$ and $V$ to calculate the fundamental group of
$X$. However, to find $pi_1(U)$ and $pi_1(V)$, I think I have to apply SVK’s theorem to each first. In order to avoid duplicate work, I will prove that
the fundamental group of CW complex $X’$ obtained from $S^1$ by attaching a two
cell with maps of degree $n$ isomorphic
to $mathbb{Z}/ nmathbb{Z}$ and apply it separately to $U$ and $V$ which are
special cases. If one lets $U’$ be some small open neighborhood of $S^1$ (and hence it contains $partial D^2$) and let $V’$ be the interior of the attached 2-cell, $U’cap V’$ is connected and
retracts to $S^1$ so it has fundamental group isomorphic to $mathbb{Z}$. Let $i_{U’}:U’cap V’ hookrightarrow U’$ be the inclusion map and $alpha$ be a
generator for $pi_1(U’cap V’)$. Then $(i_{U’})_*(alpha)=nalpha$ because the attaching map maps $partial D^2$ $n
$ times around $S^1$. The interior of $D^2$ is contractible, so we get
$$pi_1(X’)=pi_1(U’)*_{pi_1(U’cap V’)}pi_1(V’)=mathbb{Z}*{0}/ nmathbb{Z}cong mathbb{Z}/nmathbb{Z}$$

The way I understand this result geometrically is that if I have a loop that wraps around $S^1subset X’$ once, I cannot contract it to a constant loop because it wraps around $S^1$ and $S^1$ is not contractible. However, if I wrap around $S^1$ $n$ times, then the loop
traverses the boundary of the attached 2 cell $D^2$ and can“escape" (via homotopy) to the 2-cell $D^2$. $D^2$ is contractible of course, so this loop contracts to the trivial loop.

Back to the original question. My geometric arguments seems to give the fundemantal group of $X$ almost immediately. Indeed, if I have a loop that wraps around $S^1subset X$ once, it is not trivial (like before). However, if I wrap around $S^1$ twice, then my loop can "escape"to the 2-cell
attached with the map of degree 2. Thus if a loop wraps around $S^1$ 3 times, then the loop first wrapping arounds twice, which is homotopic to the identity, so it is homotopic to a loop that wraps around once. Thus attaching a
2-cell with a map of higher degree adds NOTHING to the fundamental group. More generally, if I attach 2-cells to $S^1$ with maps of varying degree, the
fundamental group is simply isomorphic
to $mathbb{Z}/ mmathbb{Z}$ where $m$ is the lowest degree of the attaching
maps. Is this intuition correct?
This
reasoning leads me to believe that the
fundamental group of $X$ is simply
$mathbb{Z}/2mathbb{Z}$. This seems
like a fairly rigorous argument to me,
but I would like to calculate directly
using SVK’s theorem.

When I intersect
$U$ and $V$, I will get a space that contracts to the $S^1$ (the 1-skeleton). Let $1$ be the generator for $pi_1(Ucap V)congmathbb{Z}$. Let $i_U:Ucap Vhookrightarrow U$ and $i_V:Ucap Vhookrightarrow V$ be the corresponding inclusion maps. Then $(i_U)_*(1)= overline{1}in mathbb{Z}/2mathbb{Z}$ and $(i_V)_*(1)= tilde{1}in mathbb{Z}/3mathbb{Z}$.
SVK’s theorem gives
$$ pi_1(X)=pi_1(U)*_{pi_1(Ucap
V)}*pi_1(V)=langle overline 1,
tilde 1| overline 1 ^2=0, tilde 1
^3=0 , overline 1=tilde
1ranglecong 0 $$
So, there seems to be a problem with my computation. How might I fix my computation? Is there a different (perhaps shorter) way of calculating $pi_1(X)$? Originally I was thinking about removing a point in the interior of each attached 2-cell and proceeding similarly to the way one would show the fundamental group of the sphere is trivial using SVK’s theorem.