
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 area enclosed by the two curves $y=\sqrt{x}$ and $y=x^2$ within the interval $x \in [0,1]$ is \_\_\_\_\_\_\_\_.
2. Given that $\tan\left(\theta + \frac{\pi}{4}\right) = -3$, find the value of $\cos 2\theta + \sin 2\theta =$ \_\_\_\_\_\_\_\_.
3. A point $A(m, 2\sqrt{2})$ lies on the parabola $y^2 = 2px (p > 0)$ with focus $F$. The chord length intercepted by the $y$-axis on the circle with center $A$ and radius $AF$ is $2\sqrt{5}$. Find the value of $m$ = \_\_\_\_\_\_\_\_.
4. Let the curve $y=x^{n+1} (n \in \mathbb{N}^*)$ have a tangent line at the point $(1,1)$ that intersects the $x$-axis at the point $(x_n, 0)$. Define $a_n = \lg x_n$. Find the value of $a_1 + a_2 + a_3 + \dots + a_{99}$ = \_\_\_\_\_\_\_\_.

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