
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
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of 2. On the interval $[-1,1)$, $f(x)$ is given by
$$
f(x) = \begin{cases}
x+a & \text{for } -1 \leq x < 0,\\
\left| \frac{2}{5} - x \right| & \text{for } 0 \leq x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left(-\frac{5}{2}\right) = f\left(\frac{9}{2}\right)$, find the value of $f(5a)$.

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