
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
An object moves in a straight line in a certain medium. At time $t$, its velocity is $v$, and its displacement is $s$. The resistance $F$ it experiences in this medium is directly proportional to the square of its velocity $v$, with a proportionality constant $k$. Given that $s= \frac{1}{2}t^2$, the work done by the object to overcome the resistance from displacement $s=0$ to displacement $s=a$ is \_\_\_\_\_\_ . (Note: The work $W$ done by a variable force $F$ is $W=\int_{s_1}^{s_2} F(s) \, ds$, and the result should be expressed in terms of $k$ and $a$.)

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