
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
Given the function $f(x) = a \sin x + b \cos x$, where $a \neq 0$, $x \in \mathbb{R}$, the maximum value of $f(x)$ is 2, and the tangent line at $x = \frac{\pi}{6}$ is parallel to the line $x - y = 0$.

1. Find the values of $a$ and $b$.
2. First, shrink the horizontal coordinates of each point on the graph of $f(x)$ to half of its original size, keeping the vertical coordinates unchanged. Then, shift the graph to the right by $\frac{\pi}{6}$ units to obtain the graph of function $g(x)$. Given that $g(\alpha + \frac{\pi}{4}) = \frac{10}{13}$, where $\alpha \in (\frac{\pi}{6}, \frac{\pi}{2})$, find the value of $\cos 2\alpha$.

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