
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 proposition p: For all $x \in [2, 4]$, $x^2 - 2x - 2a \leq 0$ always holds, and proposition q: $f(x) = x^2 - ax + 1$ is an increasing function on the interval $\left[\frac{1}{2}, +\infty\right)$. If $p \vee q$ is a true proposition and $p \wedge q$ is a false proposition, find the range of the real number $a$.

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