
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 $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $c + b\cos 2A = 2a\cos A\cos B$ $(A \leqslant B)$.  
$(1)$ Find the value of angle $A$.  
$(2)$ If the angle bisector of angle $A$ intersects $BC$ at point $D$ and $AD = 1$, find the minimum area of $\triangle ABC$.

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