
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 on a plane, establish a polar coordinate system with the origin $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. Given that point $A$ has polar coordinates $(\sqrt{2}, \frac{\pi}{4})$, and the polar equation of line $l$ is $\rho\cos (θ-\frac{\pi}{4})=a$, where point $A$ lies on line $l$.

(1) Find the value of $a$ and the Cartesian equation of line $l$;

(2) The parametric equations of curve $C$ are given by $\begin{cases} x=4+5\cos t \\ y=3+5\sin t\end{cases}$ ($t$ is the parameter). Line $l$ intersects $C$ at points $M$ and $N$, find the length of chord $|MN|$.

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