
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) Find the value of $\cos\frac{5\pi}{3}$.

(2) Given that $\frac{\sin\alpha + 2\cos\alpha}{5\cos\alpha - \sin\alpha} = \frac{5}{16}$, find the value of $\tan\alpha$.

(3) Given that $\sin\theta = \frac{1}{3}$ and $\theta \in (0, \frac{\pi}{2})$, find the value of $\tan 2\theta$.

(4) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Moreover, $a\sin A\cos C + c\sin A\cos A = \frac{1}{3}c$, $D$ is the midpoint of $AC$, and $\cos B = \frac{2\sqrt{5}}{5}$, $BD = \sqrt{26}$. Find the length of the shortest side of triangle $ABC$.

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