
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
Let $m, n, p, q$ be non-negative integers such that for all $x > 0$, the equation $\frac{(x+1)^{m}}{x^{n}} - 1 = \frac{(x+1)^{p}}{x^{q}}$ always holds. Then $\left(m^{2} + 2n + p\right)^{2q} = \quad$.

In quadrilateral $ABCD$, $\angle ABC = 135^{\circ}$, $\angle BCD = 120^{\circ}$, $AB = \sqrt{6}$, $BC = 5 - \sqrt{3}$, $CD = 6$. Then, $AD = \quad$.

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