
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 $p$: "For all $k \in \mathbb{R}$, the line $y = kx + 1$ intersects the ellipse $\frac{x^2}{a} + y^2 = 1$ at two distinct points" and $q$: "There exists an $x_0 \in \mathbb{R}$ such that the inequality $4^{x_0} - 2^{x_0} - a \leq 0$ holds." If the statement "both $p$ and $q$ are false" is false and "either $p$ or $q$ is true" is true, determine the range of the real number $a$.

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