
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 propositions $p$ and $q$:
- $p$: For all $x \in \mathbb{R}$, $x^{2}+1 \geqslant a$ holds true.
- $q$: The equation $(ρ\cos α)^{2}-(ρ\sin a)^{2}=a+2$ represents a hyperbola with foci on the $x$-axis.

Find the range of real number $a$ in the following cases:
1. If proposition $p$ is true.
2. If both propositions $p$ and $q$ are true.

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