
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
(1) Calculate the value of $(\frac{2}{3})^{0}+3\times(\frac{9}{4})^{{-\frac{1}{2}}}+(\log 4+\log 25)$.

(2) Given $\alpha \in (0,\frac{\pi }{2})$, and $2\sin^{2}\alpha - \sin \alpha \cdot \cos \alpha - 3\cos^{2}\alpha = 0$, find the value of $\frac{\sin \left( \alpha + \frac{\pi }{4} \right)}{\sin 2\alpha + \cos 2\alpha + 1}$.

(3) There are three cards, marked with $1$ and $2$, $1$ and $3$, and $2$ and $3$, respectively. Three people, A, B, and C, each take a card. A looks at B's card and says, "The number that my card and B's card have in common is not $2$." B looks at C's card and says, "The number that my card and C's card have in common is not $1$." C says, "The sum of the numbers on my card is not $5$." What is the number on A's card?

(4) Given $f\left( x \right)=x-\frac{1}{x+1}$ and $g\left( x \right)={{x}^{2}}-2ax+4$, for any ${{x}_{1}}\in \left[ 0,1 \right]$, there exists ${{x}_{2}}\in \left[ 1,2 \right]$ such that $f\left( {{x}_{1}} \right)\geqslant g\left( {{x}_{2}} \right)$. Find the minimum value of the real number $a$.

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