
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 the function $f(x) = \begin{cases} x+ \frac{1}{2}, & x\leq \frac{1}{2} \\ 2x-1, & \frac{1}{2} < x < 1 \\ x-1, & x\geq 1 \end{cases}$, and the sequence ${a_n}$ satisfies $a_1 = \frac{7}{3}$ and $a_{n+1} = f(a_n)$ for all natural numbers $n > 0$, find the value of $a_{2019}$.

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