= Solution
The two <Lasso> problems are
$$
\min_\beta\lVert Y-x\beta\rVert_2^2\quad\text{subject to }|\beta|\leq t
$$
and
$$
\min_{\beta_1,\beta_2}\lVert Y-x(\beta_1+\beta_2)\rVert_2^2
\quad\text{subject to }|\beta_1|+|\beta_2|\leq t.
$$
If $\widehat\beta$ solves the first problem, the complete solution set of the second is
$$
\boxed{\{(\beta_1,\beta_2):\beta_1+\beta_2=\widehat\beta,\
|\beta_1|+|\beta_2|\leq t\}}.
$$
Indeed, the feasible totals are exactly $[-t,t]$, the same as in the first problem.
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