
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 $q$:
- Proposition $p$: For $m \in [-1, 1]$, the inequality $a^2 - 5a - 3 \geq \sqrt{m^2 + 8}$ always holds.
- Proposition $q$: The inequality $x^2 + ax + 2 < 0$ has solutions.

If $p \vee q$ is true and $p \wedge q$ is false, find the range of possible values for $a$.

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