
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 $a$ is a positive constant, $f(x)$ is defined as:
$$
f(x)= \begin{cases} 
x^{2}-ax+1, & x \geq a, \\
x^{2}-3ax+2a^{2}+1, & x < a,
\end{cases}
$$
if there exists a $\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)$ that satisfies $f(\sin \theta) = f(\cos \theta)$, determine the range of the real number $a$.

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