
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
Given that $c > 0$ and $c \neq 1$. Let $p$ denote the statement "The function $y = c^x$ is monotonically decreasing," and let $q$ denote the statement "The function $f(x) = x^2 - 2cx + 1$ is increasing on the interval $\left(\frac{1}{2}, +\infty\right)$."

(1) If $p$ is true and $\lnot q$ (not $q$) is false, find the range of the real number $c$.
(2) If "p AND q" is false and "p OR q" is true, find the range of the real number $c$.

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