
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $f(x) = ax + b$ with $a < 0$ and $g(x) = cx + d$ with $c > 0$, both defined on $0 \le x \le 1$. Function $f(x)$ is applicable for $0 \le x \le 0.5$ and $g(x)$ is applicable for $0.5 < x \le 1$. Find the range of $r(x) = \begin{cases} 
f(x) & \text{if } 0 \le x \le 0.5 \\
g(x) & \text{if } 0.5 < x \le 1 
\end{cases}$ in terms of $a$, $b$, $c$, and $d$.

Present the answer in LaTex format: \boxed{Your answer}
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