
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 proposition $p$ be: the function $y=c^x$ is strictly decreasing on $\mathbb{R}$; and proposition $q$ be: the function $f(x)=x^2-2cx+1$ is increasing on $(\frac{1}{2},+\infty)$. If the proposition $p \wedge q$ is false, and the proposition $p \vee q$ is true, find the range of values for the real number $c$.

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