c^2 = a^2 + b^2 - 2abcosC ②

Add and simplify to obtain

a = ccosB + bcosC

Combined with the known 3acosA = bcosC+ccosB.

get:3a cosa = a = = >； cosA = 1/3

(2) According to COSA =1/3 = = = > Sina = √8/3

cosB =-cos(A+C)=-cosa cosc+Sina sinC =- 1/3 cosc+√8/3 * sinC③

It is also known that cosB+cosC = √ 12/3 is substituted into ③.

cosC + √2sinC = √3

√ (1-sinc 2) = √ 3-√ 2 sine

SinC = 2/√6 = √6/3。

It is known that A = 1.

Sine theorem: c = asinc/Sina =1* √ 6/3/(√ 8/3) = √ 3/2.

Answer: c =√3/2

This problem is relatively simple, but the amount of calculation is a bit large.