Pareto allocations and Competitive equilibrium

Consider the one-consumer one-firm economy.
The consumer has preferences over leisure $lin(0,L)$ and consumption good $x ≥ 0$ represented by utility function $u(x, l) = ax + l$, where $a > 0$ is a parameter. Moreover, the consumer is endowed with $(0, L)$, that is, has zero units of the consumption good and $L > 0$ units of leisure as an endowment.

The firm uses the production function $f: R_{geq0}rightarrow R$ to produce the consumption good out of labor. If the firm uses $z ≥ 0$ units of labor it produces $f(z)$ units of the consumption good, where:

$$f(z)=begin{cases}
0 &text{if } z=0
z+1 &text{if } zin(0,L)
L &text{if } zgeq L
end{cases}$$

The price of the consumption good is p and the price of labor is $w$. The
consumer owns the firm.

The question asks for the Pareto efficient allocations in the case where $a=1$

I guess we need to solve for $$text{max} u(x,l) text{s.t} xleq f(L-l) $$
and we should consider 3 cases.

When $l=L$ we have $f(L-l)=0$ and $u(0, L) = L$

When $l = 0$ we have $f(L − l) = L$ and $u(L, 0) = L$

When $l in (0, L)$ we have $f(L − l) = L − l + 1$ and $(L − l + 1, l) = L + 1$

But based on these how can we state the PO allocations? and if we had that $aneq1$ what would be the competitive equilibria in this case?

Because when again $a=1$ profit allocations would be stated as :

when $z=0$ $pi=p*0-w*0=0$

when $zin(0,L)$ $pi=p*(z+1)-wz=z(p-w)+p$

when $zgeq$ $pi=L(p-w)$ only in the case when $z=L$ otherwise it won’t be optimal.

and we need to solve the consumer’s utility maximization problem as well in order to find CE

But I’m stuck with the other case when $aneq1$