
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
In the Cartesian coordinate system $xoy$, with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho = 2\cos \theta$. The line $l: \begin{cases} & x=\frac{1}{2}t+a \\ & y=\frac{\sqrt{2}}{4}t \end{cases}$ (where $t$ is a parameter) intersects curve $C$ at points $A$ and $B$.

$(1)$ Find the range of the real number $a$;

$(2)$ Let point $M(a,0)$, if $|MA| \cdot |MB| = \frac{5}{4}$, find the value of the real number $a$.

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