
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
Points $M$ and $N$ belong to the sides $AB$ and $AC$ respectively of the isosceles triangle $ABC$, with $MN \parallel BC$, and a circle can be inscribed in the trapezoid $BMNC$. The radius of this inscribed circle is $R$, and the radius of the inscribed circle of triangle $AMN$ is $r$. Find

a) the base $BC$

b) the distance from point $A$ to the nearest point of tangency;

c) the distance between the chords of the circles that connect the points of tangency to the lateral sides of the trapezoid $BMNC$.

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