
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 that the function $f(x)$ is an even function defined on $R$, and when $x \geqslant 0$, $f(x) = \dfrac{x-1}{x+2}$, determine the range of real number $a$ if the inequality $f(2t-1) < f\left(\dfrac{1}{3}a\right)$ holds for any real number $t \in \left[-\dfrac{1}{4}, \dfrac{1}{2}\right)$, where $a > 0$.

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