
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 $c > 0$, let $p$ be the proposition that the function $y = \lg[(1-c)x - 1]$ is an increasing function within its domain, and $q$ be the proposition that the solution set of the inequality $x + |x - 2c| > 1$ is $\mathbb{R}$. If the proposition "$p \vee q$" is true and "$p \wedge q$" is false, find the range of the real number $c$.

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