
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
Find the value of the following expressions:
\[
(1)\ \lg 5^{2}+ \frac{2}{3}\lg 8+ \lg 5 \cdot \lg 20 + (\lg 2)^{2}
\]
\[
(2)\ \cos \frac{17\pi}{4} + \sin \frac{13\pi}{3}+ \tan \frac{25\pi}{6}.
\]

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