
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) Given that $\tan(\alpha - \frac{\pi}{4}) = 2$, find the value of $\cos(2\alpha)$.

(2) Given the moving straight line $(2 + \lambda)x + (1 - 2\lambda)y + 4 - 3\lambda = 0$ intersects with the circle $C: (x - 1)^2 + y^2 = 9$, find the length of the shortest chord of intersection.

(3) Given vectors $a$ and $b$ such that $|a| = 3$, $|a - b| = \sqrt{13}$, and $a \cdot b = 6$, find the measure of the angle between $a$ and $b$.

(4) A bus station has buses departing at exactly $6:30$, $7:00$, and $7:30$ in the morning. A student arrives at the station between $6:50$ and $7:30$ to catch a bus, with the arrival time being random. Find the probability that the student waits no more than $8$ minutes for a bus.

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