
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 an ellipse $M$ with the right focus at $F(1,0)$ and passing through the point $D(1, \frac{3}{2})$, described by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$):

$(1)$ Find the equation of the ellipse $M$;  
$(2)$ A line through point $F$ intersects the ellipse $M$ at points $A$ and $B$. If the slopes of lines $DA$, $DC$, and $DB$ form an arithmetic sequence, find the maximum value of $\tan \angle DCF$.

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