
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
Calculate the area enclosed by the following lines:

1) The parabola \(4y = 8x - x^{2}\) and the line \(4y = x + 6\);

2) The parabolas \(y = 4 - x^{2}\) and \(y = x^{2} - 2x\);

3) The cubic parabolas \(6x = y^{3} - 16y\) and \(24x = y^{3} - 16y\);

4) The ellipse \(x = a \cos t, y = a \sin t\);

5) The cardioid \(\rho = a (1 + \cos \varphi)\);

6) The circles \(\rho = 2 \sqrt{3} a \cos \varphi\) and \(\rho = 2a \sin \varphi\).

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