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

(2) Calculate the value of $\left( \frac{1}{2}\right)^{-2}+\lg 2-\lg \frac{1}{5}$.

(3) The ratio of the number of students in the first, second, and third grades of a high school is $5:4:3$. If a stratified sampling method is used to draw a sample of size $240$, then the number of second-grade students drawn is ____.

(4) If the function $f(x)=\lg ({{10}^{x}}+1)+ax$ is an even function, then the real number $a=$____.

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