
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: $x_1$ and $x_2$ are the two real roots of the equation $x^2 - mx - 2 = 0$, the inequality $a^2 - 5a - 3 \geq |x_1 - x_2|$ holds for any real number $m \in [-1,1]$; Proposition Q: There is only one real number $x$ that satisfies the inequality $x^2 + 2\sqrt{2}ax + 11a \leq 0$. If proposition P is false and proposition Q is true, find the range of values for $a$.

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