
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
(1) The graph of the function $y = \log_a(x + 4) - 1$, where $a > 0$ and $a \neq 1$, always passes through a fixed point $A$. If point $A$ lies on the line $mx + ny + 1 = 0$, where $m, n > 0$, find the minimum value of $\frac{1}{m} + \frac{2}{n}$.

(2) The function $f(x)$ satisfies the equation $f(x) = x^2 + 3f'(1)x - f(1)$. Find $f(4)$.

(3) The geometric sequence $\{a_n\}$ satisfies the equations $a_1 + a_3 = 10$ and $a_2 + a_4 = 5$. Find the maximum value of $a_1a_2 \cdots a_n$.

(4) The base $ABCD$ of a square pyramid $P-ABCD$ is a square with side length $AB = 2$. The height $PA$ is perpendicular to the base. If all the vertices of the pyramid lie on a sphere with a surface area of $16\pi$, find the length of $PA$.

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